Educational Studies in Mathematics - Volume 51

ADVICE TO PROSPECTIVE AUTHORS Educational Studies in Mathematics is an international journal of research and scholarshi...

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ADVICE TO PROSPECTIVE AUTHORS

Educational Studies in Mathematics is an international journal of research and scholarship in mathematics education. It aims to illuminate issues of principle, policy and practice in the field, and to promote the development of coherent bodies of theorised knowledge which can be brought to bear on these issues. Its intended readership includes those involved in teaching and studying mathematics education and those with responsibilities for professional leadership, as well as active researchers and scholars in the field. The journal seeks to publish articles that are clearly educational studies in mathematics, make original and substantial contributions to the field, are accessible and interesting to an international and diverse readership, provide a well founded and cogently argued analysis on the basis of an explicit theoretical and methodological framework, and take appropriate account of the previous scholarly work on the addressed issues. Submissions that demonstrate all these qualities are at a considerable advantage. Authors are advised to make the relevance and significance of their contribution clear in the introduction to their article and its conclusion. Although the work reported in a submission may well be located in a particular local or national context, it should be of interest to a wider audience, and written up in a way which makes it comprehensible to them. The editors recognise that many different forms of research and scholarship can contribute to the aims of the journal, and that these will draw on differing perspectives and approaches. But, in treating a particular area or aspect of mathematics, a submission should show critical awareness of other possible approaches. In particular, authors will be expected to be familiar with work already published in the journal, and to acknowledge, or build on it as appropriate. Equally, it can help readers who may be relatively unfamiliar with the topic and wish to learn more, if some reference is made to appropriate published sources which offer an authoritative overview of the area under consideration. Whatever approach a submission adopts to evidence and argument, it will be evaluated in terms of appropriate criteria of rigour, intended to ensure that the analysis is well founded and that it develops a resourceful and convincing argument. In engaging directly with a body of evidence, a submission should make explicit the Educational Studies in Mathematics 51: 1–2, 2002.

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theoretical and methodological framework within which this evidence has been gathered and analysed. Both editors and readers of the journal appreciate conciseness, as long as it is not at the expense of clarity. Articles exceeding 20 journal pages in length are unlikely to be published unless they are of unusual significance and an extended presentation is clearly necessary to do justice to the material. Submissions that successfully distil the essence of an argument into 10 pages rather than 20 are particularly welcome, as this both focuses the particular communication and enables the journal to make a greater range of material available to its readership. T HE E DITORS

JO BOALER

EXPLORING THE NATURE OF MATHEMATICAL ACTIVITY: USING THEORY, RESEARCH AND ‘WORKING HYPOTHESES’ TO BROADEN CONCEPTIONS OF MATHEMATICS KNOWING1

ABSTRACT. What proficiencies are brought to bear when students work on mathematics problems? And to what extent may these be captured by knowledge categories? These are questions that I consider in this article, as I explore notions of competency, that go beyond knowledge to include the mathematical ‘dispositions’ that students bring to problems and the ‘practices’ with which they engage. This exploration will draw from two frameworks that have recently been introduced in the US. In addition, I consider the ways in which research knowledge is conceived and developed, reflecting upon the important role of theory and the potential of ‘working hypotheses’ for connecting with practice in new ways. KEY WORDS: dispositions, knowledge, mathematical practices, research, theory and practice

1. I NTRODUCTION

It is some years now since the day I received an important envelope in my mailbox in London, carrying the ESM postmark. I opened it nervously, knowing that the contents would tell me whether the first article I had written for publication, had been accepted. To my delight it was and that time marked not only my own beginning relationship with the journal, but with the scholarship of mathematics education. I was a masters student at the time, and unsure whether I could contribute to the world of academia. Educational Studies in Mathematics 51: 3–21, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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I had submitted my paper to ESM, on the advice of one of my university professors, who told me it was one of the most important journals in mathematics education. Publishing in ESM was a wonderful way to begin my writing career, as the editor at that time, Leen Streefland, was supportive and encouraging in helping me prepare my article for publication. Since then ESM has continued to play an important role in my work, providing a forum through which I learn from research and ideas produced all around the world; indeed its international scope is one of the aspects I most appreciate about the journal. It was with great pleasure then that I received an invitation to write this paper, as part of the celebration of the 50th anniversary of the journal. I was asked to do so, as a current user of the journal, someone who publishes in ESM and uses it in teaching and research. I have therefore chosen to reflect upon some ideas that I am working on now, that ESM has helped stimulate, problematise and nourish, through articles that have appeared. I will consider in this short paper, what it means to have a broad conception of knowing – for research and for mathematics – reflecting upon the contribution of a selection of ESM articles as I do so.

2. U SING THEORY TO ADVANCE KNOWLEDGE

The task of researchers and scholars in mathematics education, as in other academic fields, is to create new knowledge. This is no small endeavour and it carries with it critical responsibility to be open to new ideas, perspectives and ways of thinking. For if scholars do not entertain broad ways of thinking, then scholarship is in danger of closing in, and of moving only in concentric, or ever decreasing circles. Theory is critical to the production of research knowledge, and to educational work more generally. Researchers in mathematics education have been receptive to the important role played by theory, with many classic studies drawing upon theoretical frameworks carefully; combining different theories from within and outside education; and even contributing to the development of new theories (see for example, Dubinsky, Sfard, Tall). But careful use of theory is not a hallmark of all educational research and the ways in which theory may be used to enhance our knowledge of mathematics teaching and learning seems worthy of some careful reflection and explicit teaching (Boaler, Ball and Even, 2003). Stephen Ball contends that the value of theory is that it: can separate us from ‘the contingency that has made us what we are’ (. . .) Theory is a vehicle for ‘thinking otherwise’ (. . .) It offers a language for challenge, and modes of thought, other than those articulated for us by dominant others. It provides a language of rigour and irony rather than contingency. The purpose of

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such theory is to de-familiarise present practices and categories, to make them seem less self evident and necessary, and to open up spaces for the invention of new forms of experience. (Ball, 1995, p. 266)

One of the roles of theory, as Ball notes, is to protect us from the limits of our own experience, which inevitably narrow our thinking. Theoretical frameworks encourage researchers to pursue new ideas, prompting selections and explorations of data that our experiences would not initiate. Theory also serves as an interpretive tool, helping researchers to understand particular interactions that take place, by positioning them – providing dimensions along which ideas may be located and examined. McDermott and Lave (2002), like Ball, urge the careful use of theory as a way of prevailing against ideological subjection and dogma: We cannot trust ourselves to think our way to ideas that we need to change our lives. We need help. One kind of help is to work on rich texts that force us systematically to relocate our work with the work of others. (McDermott and Lave, 2002, p. 46)

But researchers in mathematics education do not only need to use theory, they need to select theoretical frameworks carefully. McDermott and Lave allude to the importance of theory in tackling big issues of social justice and equality, in producing work that may ‘change lives’ (p. 46). They argue that narrow frameworks may not give us the perspective to question prevailing practices with which we are familiar and which it is hard to see beyond. In 1990 Yves Chevallard took up a similar issue in ESM, criticizing research in mathematics education for its narrowness. He claimed that: research in mathematics education must certainly broaden its outlook, and take into account determinants which it has so far flippantly ignored. It is also its duty, nevertheless, to investigate patiently, even punctiliously, the relationship between the individuals’ experience and conduct and the socially determined contexts in which they emerge. (Chevallard, 1990, p. 24)

Chevallard spoke to the importance of broadening the questions mathematics educators had been asking, the variables considered, and the frameworks employed, as well as drawing connections between different frameworks. Whereas Ball warned against the dangers of becoming atheoretical, Chevallard’s article reminded us that the singular adoption of particular theories or frameworks can also be narrowing. The vast majority of early inquiries in mathematics education drew from the same, or similar perspectives and frameworks, and many would acknowledge that this helped our field to gain strength and to develop a clear identity. It also allowed knowledge to cumulate in careful ways. Researchers who studied students’ mathematical conceptions, for example,

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worked collectively across a number of countries to map out progressions of subject understanding in a few key areas (for example Hart, 1981; Dickson, Brown and Gibson, 1984). But the field has now diversified (Lerman, 2000), and researchers draw from many different methodologies and frameworks. This has prompted some crises in identity (Sierpinska and Kilpatrick, 1998), and increased demands for new researchers entering our field who need to learn about previous work, but it also paves the way for many new and important inquiries in the future of mathematics education. This point was well illustrated in an article that appeared in ESM in the mid nineteen-nineties. At a time when constructivism had become a dominant paradigm in mathematics education, displacing previous theories of learning, Robyn Zevenbergen (1996) wrote an article in which she questioned the limits of constructivism, claiming that it had attained an unhealthy dominance within the field. She was particularly concerned about its inability to “theorise adequately the marginalisation of significant numbers of students” (1996, p. 96). In this important essay Zevenbergen illustrated the role of theoretical breadth, of the use of multiple theories to counter dominant hegemonies. One of Zevenbergen’s main propositions was that success in mathematics is not simply a matter of cognitive processes, and that “students from certain social and cultural groups are more likely to be constructed as effective learners of mathematics because of their congruency with the social context of formal schooling” (p. 105). Zevenbergen drew from sociological theory to argue this powerful idea, encouraging researchers to locate understandings of students’ engagement and success within a broad sphere, that extended beyond students’ interactions with curricular materials. This idea emerged through the careful juxtaposition of different theories, including constructivism and cultural capital (Bourdieu, 1982, 1986), as Zevenbergen explored their differences and the insights they gave. Ball, Chevallard, Lave, McDermott, and Zevenbergen, all concern themselves with an extremely important issue – that of breadth of thought and of openness. They remind us that theoretical perspectives play an important role, prevailing against ideology and dogma. They also remind us that theories must be employed with care and reflectivity, in order to preclude a form of narrowness that comes from the unquestioning acceptance of dominant paradigms. Chevallard and Zevenbergen urge mathematics educators to look beyond single frameworks, remaining open to the different ways that theories from within and outside mathematics education may illuminate some pressing and enduring questions in our field, such as those of social inequality. The different authors speak to the importance of a

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research knowledge that is broadly conceived, enhanced by theoretical reflexivity. But what does it mean to be aware of diverse epistemological perspectives, to show appreciation for them and to draw upon them? If one surveys the landscape of mathematics education research it is possible to find many examples of scholars who have embraced different epistemological positions and produced important new knowledge because of such openness. In the past, mathematics education researchers drew largely from psychological frameworks and theories, but contemporary researchers are increasingly demonstrating the insights that may be learned from additional frameworks. Central among the frameworks now drawn upon are sociology (Ensor, 2001; Dowling, 1996; Morgan, 2000); sociocultural theory (Kieran, Forman and Sfard, 2001; Lerman, 2001); politics (Skovsmose, 1994; Valero, 1999; Vithal and Skovsmose, 1997); mathematics (Ball and Bass, 2000); philosophy (Ernest, 1991, 1999); history (Joseph, 1992) and anthropology (Chevallard 1992; Artigue, 1999). In these different accounts the researchers draw from varied disciplinary perspectives and make use of methodological frameworks that are associated with them. But scholarship is anything but simple, and whilst we may agree that breadth of thinking is critical to the evolution of ideas, and that different frameworks should be considered and employed, we must also be wary that mathematics education is a relatively new and young field and that too much breadth and diversity will cause a scattering of focus and preclude opportunities for consolidation and identity (Sierpinska and Kilpatrick, 1998). Indeed work that starts from the findings of previous research, and builds from such findings, to provide depth and texture, such as the history of work on misconceptions and cognitive change contributes significantly to our understandings. Over time we must hope that journals continue to support both types of work, as our field diversifies and grows. Indeed research journals, such as ESM, play an extremely important role in providing an avenue of communication for different theories, as well as arguments about the use of theory. Over the years ESM has continued to encourage breadth and openness of thought through the publication of articles such as Zevenbergen’s that challenge dominant ways of thinking, as well as through its encouragement of different genres of article; different modes of research; and different frameworks of analysis. 3. R ELATIONS OF KNOWLEDGE AND PRACTICE

The ways in which theory may support work in the scholarship of mathematics education is an important question for our field, but an equally

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important question to consider is the ways in which theory may impact the practices of mathematics teaching and learning. Educational researchers, unlike some other scholars, are working in an applied field and our work is judged, in part, by the extent to which it is able to impact – indeed improve – educational practice. It is in this area that scholarship in education is most often found to be lacking. Greeno, McDermott, Cole, Engle, Goldman, Knudsen, Lauman, and Linde (1999) provide an interesting perspective on this issue. Greeno and his colleagues suggest a new conceptualisation for knowledge and theory in education. They provocatively claim that we remove the boundaries between knowledge and “domains of practical activity” (1999, p. 303) suggesting that a means of doing so would be to engage teachers, researchers and students in new participation structures, in which they work together to produce new research knowledge. Such structures are not, themselves, provocative, but Greeno et al. propose that “expert knowledge is better seen as a working hypothesis that must enter a community of practice and jostle apparent knowledge until it takes root in a reorganization of what people can do with each other” (1999, pp. 301–302). Drawing from both Dewey (1916) and Mead (1899), they offer a different conceptualisation of knowledge as a “set of working hypotheses that would be tested against their consequences for the body politic” (p. 333). Thus some of the knowledge produced by educational scholars should be judged against the extent to which it impacts practice. This view is echoed in a National Academy of Education report (NAE, 1999) from the US that suggests that educational research, in future, should be judged not only on its appeal to other academics, or its potential to improve education, but on its impact. The report proposes a change for educational funding, with priority given to those studies that aim to impact practice, on which researchers and practitioners work collectively to frame problems of practice, and their solutions. The National Academy of Education group recommends that educational research is reorganized with new kinds of support, review, participation and evaluation, all organized to enable the collaboration of different ‘communities’ such as researchers, practitioners and developers of materials. They support a kind of research that they call ‘problem solving research and development’ – characterised by researchers and practitioners identifying problems of practice together and working together to solve them. In doing so they challenge an assumption that research knowledge should be developed as general principles that should then be conveyed to teachers and other education professionals. They replace this vision of research and dissemination with one in which the development of knowledge, understanding and educational improvement takes place as one

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process, shared by specialists in research, development and educational practice. The proposal to re-conceptualise knowledge as a set of working hypotheses, tested against their ability to impact practice, is an interesting one for a journal such as ESM. Greeno et al. do not suggest that such knowledge should replace discipline based knowledge that is valued for its own sake, but they argue for an expansion of the ways research knowledge is developed and valued in our field. I began this article with discussion of the ways knowledge may be enriched by theory – this is another call for the enrichment of knowledge, this time by practice, and it raises an interesting question for journals such as ESM. Perhaps now, on the anniversary of its 50th birthday, at a juncture where there are serious dichotomies between research knowledge and the practices of education, it is time for ESM to give its authors new directions that could change the way research knowledge is developed and used. These could include instructions for authors to reflect upon the ways their research findings may be used in different settings, and the means by which they will be communicated to, and taken up by practitioners to improve students’ educational opportunities. This does not mean that new knowledge would not be valued for its own sake, or that there would be no place for conceptual essays in ESM. But perhaps researchers could be urged to consider their findings as ‘working hypotheses’ and explore the ways in which their knowledge may enter different communities and “jostle apparent knowledge until it takes root in a reorganization of what people can do with each other” (Greeno et al., 1999, pp. 301–302)? Whether this is feasible, desirable, or possible, is a question that seems worthy of consideration. The expansion of research knowledge – to benefit fully from theory, and to connect in new ways with practice, is an issue that has been furthered by a few important contributions in ESM, that call for a prevailing openness in our conceptions of knowledge. But it is knowledge of mathematics itself, rather than research knowledge, that has gained the greatest attention of ESM authors. Indeed the question of what it means to know mathematics, and be proficient in its use, is one to which researchers of mathematics education have contributed a great deal. I will end this paper with some consideration of the domain of mathematics itself, as explorations of the ways mathematics knowledge may be opened, expanded, and enriched have an important role to play in our work as researchers of mathematics education, as I shall argue below.

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4. E XPANDING CONCEPTIONS OF MATHEMATICAL KNOWING AND DOING

Many articles over the years have reported on research that has considered how mathematics is learned – which approaches are effective, for whom and why. This work is at the centre of our field and is extremely important. A related question, that has also received considerable attention, concerns the nature of mathematics knowledge – what is it, and how is it, or could it, be held? I have spent time in this article considering research knowledge and the ways it may be expanded by the contributions of theory and practice. I turn now to the question of mathematics knowledge, and consider the ways that recent work may benefit our conceptions of knowledge, expanding and opening these to encourage insights into the relations of mathematics knowing, doing and believing. Ball, McDermott, Lave, and others all stress the importance of thinking openly about educational issues, using theory to protect against narrow interpretations constrained by personal experience. The issue of openness is also critically important for those who consider the nature of mathematics knowledge, particularly because the prevailing dogma about what it means to know and be proficient in mathematics is extremely narrow in most countries (Boaler, 2002a). Indeed, one could argue that it is the narrowness with which mathematics is regarded that has maintained a system of educational failure, in which only a few ever attain mathematical proficiency or fluency. In the two countries in which I have worked – the UK and the US – mathematics is believed by many students to be a collection of disconnected, standard procedures (Boaler, 2002a; Schoenfeld, 1992). In many schools, homes, and departments of education, test success is held as the ultimate goal, and students of mathematics often come to believe that their goal is to memorise numerous different, unrelated procedures, so that they can reproduce them when they are given different test questions. This scenario can lead to a limited test-knowledge, or worse, as students often fail even to be successful on tests, as they find that memorised procedures are insufficient when faced with questions that require that they also know when different procedures should be selected (Brown, 1981; Boaler, 1997, 2002a). Research in mathematics education has contributed a great deal to our understanding of these problems, and various authors in ESM have been particularly helpful in exploring the boundaries of mathematics knowing. In the nineteen-seventies and eighties there was important work, and agreement about different forms of knowledge that may be developed, that have been variously characterised as conceptual and procedural (Hiebert,

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1986), or relational and instrumental (Skemp, 1976). These characterisations of knowledge have been important, and they have resonated with many teachers and scholars. But recent articles in ESM capture some different, important aspects of mathematical work that seem to elude a knowledge characterisation. Indeed when we step outside of knowledge domains, it seems that our field may lack agreed frameworks for understanding mathematical activity. As an example consider an excellent article by Noss, Healy, and Hoyles (1997). In this research paper authors describe the ways in which they developed a particular ‘microworld’, which was designed to “help students construct mathematical meanings by forging links between the rhythms of their actions and the corresponding symbolic representations they developed” (p. 203). In this article Noss, Healy, and Hoyles examine the ways in which students make connections between visual and symbolic forms of functional relationships. They argue, importantly, that students often develop a disconnected sense of algebraic formulations – regarding algebra as an end-point, rather than a problemsolving tool. The authors therefore designed a computer environment in which the only way to manipulate and reconstruct objects was to explicitly express the relationships between them. In doing so they helped the students view and use algebra as a representational tool, in the service of the expression of mathematical connections and relationships. Noss, Healy, and Hoyles (1997) argue that the act of making connections is important because mathematical meanings derive from mathematical connections. Thus they link mathematical connections to the knowledge and understanding they may promote. But it seems to me that the act of observing relationships and drawing connections, whether between different functional representations or mathematical areas, is a key aspect of mathematical work, in itself, and should not only be thought of as a route to other knowledge. In Leone Burton’s article of 1999, she reported upon a study in which she interviewed 70 research mathematicians, probing the ways that they viewed and used mathematics. One of the key aspects of their work that the mathematicians highlighted was also the act of making connections – as one of them reflected:

The behaviour that I observed I couldn’t find anything about it in the standard literature. Then I found a connection with a very ancient problem in gravitational mechanics and I found some old computational work from the 1960’s and the behaviour they found was almost identical but even richer than the behaviour I was finding. So whilst I was understanding more and more about my problem, I was also seeing how it was linking into this huge area of the 3-body problem . . . That was really nice. I think that is pretty common in maths. Things do connect when you don’t expect them to. (Male lecturer). (Burton, 1999, p. 136)

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Andrew Wiles achieved one of the most significant mathematical breakthroughs of our time in his proof of Fermat’s Last Theorem. Biographical accounts of his work (e.g. Singh, 1998) highlight the significance of the connections Wiles was able to draw between different mathematical theories; indeed it was the connections that he and other mathematicians drew that laid the path for the eventual proof. Frey, a German mathematician, set the ball rolling when he claimed that “if anyone could prove the TaniyamaShimura conjecture then they would also immediately prove Fermat’s Last Theorem”. Ken Ribet and other mathematician’s worked hard to complete the connection between the Taniyama-Shimura conjecture and Fermat’s Last Theorem, but could only prove “a very minor part of it” (Singh, p. 201). But, as Singh recalls: Fermat’s Last Theorem was inextricably linked to the Taniyama-Shimura conjecture. (. . .) For three and a half centuries Fermat’s Last Theorem had been an isolated problem, a curious and impossible riddle on the edge of mathematics. Now Ken Ribet, inspired by Gerhard Frey, had brought it center stage. The most important problem from the seventeenth century was coupled to the most significant problem of the twentieth century. A puzzle of enormous historical and emotional importance was linked to a conjecture that could revolutionize modern mathematics. (Singh, 1998, p. 202).

Wiles subsequently went on to prove the Taniyama-Shimura conjecture, thereby proving Fermat’s Last Theorem, a breakthrough that derived from the connections drawn. Burton and Singh both write about the work of mathematicians, focusing upon the act of ‘making connections’. But what is this aspect of mathematical work? And has its character – as an action or mathematical practice – rather than a form of knowledge or knowing – contributed to its relative lack of attention in curriculum materials and teaching? For the act of making connections is not something students need to know, it is something they need to do. One could imagine a student with a broad knowledge of mathematical procedures and even a conceptual understanding of the relations between procedures, who still would not choose to draw connections between different mathematical ideas, relations or representations as they work. Students may think of mathematics as a set of connected ideas, they may even be able to describe and reflect upon the connections, but the act of linking one mathematical area to another, in order to solve a problem, is an action that extends beyond knowledge. Students with a deep understanding may draw connections as they work, as knowledge and practice are intricately connected, but the act of doing so is not defined by the knowledge they possess. This raises the question of whether mathematics education researchers have focused too centrally upon knowledge categories, neglecting various mathematical actions, such

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as those to which Burton; and Noss, Healy, and Hoyles, draw our attention, that are so critical to mathematics work. Before pursuing this question further I will consider another example of research, published in ESM, that highlights another aspect of mathematical work that eludes knowledge categorisation. In 1996 Marty Simon drew our attention to what he calls a mathematical ‘ability’ – that of considering a mathematical problem, with the various constraints and variables described, not as a static state, but a dynamic process. Simon gives as an example a girl who is considering whether she can make an isosceles triangle if only two angles and the included side are specified. The girl immediately represents such a triangle and justifies it saying “Well, I know that if two people walked from the ends of this side at equal angles towards each other, when they meet, they would have walked the same distance” (1996, p. 199). Simon argues that such reasoning is not inductive – the girl did not generate several triangles and notice a pattern. Nor is the reasoning deductive – she did not make a conjecture and create a need for a deductive proof. Rather she saw the isosceles triangle not as a static figure with particular dimensions, but a “dynamic process that generates triangles from the two ends of a line segment”. This dynamic mental model, as Simon points out, enabled her to reason about two ideas, that she connected – the relationship between the base angles of a triangle, and the relative length of the legs of the triangle given particular angles. Simon describes this mode of work as “seeking a sense of how the mathematical system works” or developing a “feel for the system” (p. 198) and refers to this process as “transformational reasoning” (1996, p. 197). Simon, like Burton, Noss, Healy, and Hoyles, describes a particular action – of viewing mathematical relationships as a dynamic state. This action, like that of noting relationships or drawing connections, also eludes knowledge characterisation, and Simon refers to it as a particular type of reasoning. Ben-Zvi and Arcavi (2001) analyze the work of students who were learning to use exploratory data analysis in a technological environment. In their descriptions of the students working Ben-Zvi and Arcavi highlight the importance of what they call ‘habits’ such as questioning, representing, concluding, and communicating. Ben-Zvi and Arcavi stress the importance of enculturation as an act of teaching, with teachers inducting students into statistical work, so that they may learn a variety of ‘thinking processes’ and ‘problem solving strategies’. Their focus in this paper also extends beyond the knowledge that students need to explore data, to some important mathematical actions, such as “looking globally at a graph as a way to discern patterns and generalities” (2001, p. 38).

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The different research projects I have considered all describe critical mathematical actions that extend beyond knowledge – but when we consider how such actions are characterised or defined, things look a little hazy. Some authors have offered broader notions of knowledge arguing that these different actions are indicative of different forms of knowledge. In a special issue in 1999, for example, (Tirosh, 1999), a range of authors described different categories of knowledge such as implicit, explicit, formal and visual knowledge, as well as knowing that, knowing why and knowing how. These notions extend traditional conceptions of knowledge, enabling researchers to account for broader aspects of competent performance, but it seems that the different actions people employ are not defined by this knowledge. Other scholars have described the kind of actions I have reviewed as mathematical processes, with names such as reasoning, and communicating (Schoenfeld, 1985, 1992; NCTM, 2000). At this time there is little agreement about the nature, or form of mathematical actions, such as visualising, or connecting, and this may be part of the reason that these critical aspects of mathematics work are frequently overlooked in curriculum planning and assessment. In these different characterisations of the mathematical work employed by students and mathematicians we also gain an important sense of some mathematical traits that supported the work, including creativity, interest, and inquisitiveness. Indeed it is hard to believe that such characteristics could be separated from the work of drawing connections or regarding relations dynamically. Yet these characters are similarly elusive and difficult to define – sometimes being considered as mathematical ‘beliefs’, at other times ‘habits of mind’ (Cuoco, Goldenberg and Mark, 1996). Such habits, or beliefs are also given little attention when policy makers and schools make decisions about curriculum materials, and teaching strategies. This lack of attention, I would argue, has been encouraged by the separate study, and labelling, of characteristics such as mathematical knowledge, ‘processes’ and ‘habits of mind’. Thus scholars and teachers have often focused upon knowledge as though it develops independently of belief, action, or disposition. This is particularly true of studies of students’ knowledge – scholars of teacher knowledge have focused to a greater extent upon the intrinsic relationships between teacher knowledge and belief (see, for example, Even and Tirosh, 1995; Cooney, 1999; Thompson, 1992) whereas it is still relatively commonplace for studies of student learning to focus only upon knowledge developed. Two groups in the United States, comprising mathematics educators, teachers, mathematicians and policy makers, recently produced different conceptualisations of what it means to know and use mathematics that

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seem helpful in furthering our understanding of the different relations of mathematical knowledge, beliefs and actions. First the National Research Council (NRC) convened a group of experts, led by Jeremy Kilpatrick, to review and synthesize relevant research on mathematics learning, from pre-kindergarten to the end of grade eight (Kilpatrick, Swafford and Findell, 2001; for a concise account of this work see Kilpatrick, 2001). As part of this work, the group provided a conceptualisation of successful mathematics learning, that they called ‘mathematical proficiency’. The different, interwoven, aspects of proficiency they proposed, are: conceptual understanding – comprehension of mathematical concepts, operations and relations procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately strategic competence – ability to formulate, represent and solve mathematical problems adaptive reasoning – capacity for logical thought, reflection, explanation, and justification productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (Kilpatrick et al., 2001, p. 116)

This representation of proficiency seems important, particularly in its inclusiveness, and its bringing together of knowledge, aspects of mathematical work, and disposition. Practices such as making connections, and viewing mathematical representations dynamically, could be included as examples of ‘strategic competence’, whereas characteristics such as creativity, and inquisitiveness may be included under ‘productive disposition’. The utility of this framework will become clearer in time, as researchers, teachers and others work with the ideas, but it seems that it has the potential to do something very important – expand public conceptions of mathematics knowing and turn attention to the aspects of mathematical proficiency that need to accompany knowledge. A second group of mathematicians and educators in the US, led by Deborah Loewenberg Ball, recently designed a proposal for future research directions in mathematics education. This proposal (RAND mathematics study panel, 2002) is intended to guide government agencies in their distribution of research funds. The proposal has many interesting features, including a call for strategic accumulation of research in focused areas, and collaborations of different groups working on problems together. Most interesting for this discussion is the choice of one of the three proposed research directions for the future as that of ‘mathematical practices’. The group recommends this area as a way of adding texture and understanding

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to the notion of mathematical proficiency, and they describe mathematical practices in the following way: This area focuses on the mathematical know-how, beyond content knowledge, that characterizes expertise in learning and using mathematics. The term ‘practices’ refers to specific things that successful mathematics learners and users do. Justifying claims, using symbolic notation efficiently, and making generalizations are examples of mathematical practices. (RAND, 2002, p. xi)

The group, of which I was a part, was in agreement that the field would be significantly advanced by focused work on these practices – considering what they are, how they are learned, and how they are used, in the service of employment, recreation and mathematical inquiry. Examples of mathematical practices that the group highlighted for study include justification, representation, and reconciliation. The actions of making connections (Noss, Healy, Hoyles, 1997; Burton, 1999), and seeing mathematical relations as a dynamic process (Simon, 1996) could also be taken as candidates for study in this area. The notion of practices, central to situated theory (Lave, 1993; Wenger, 1998), has been extremely generative in mathematics education, as researchers have begun to look beyond students’ cognitive processes, to the norms of classrooms (Cobb, Wood and Yackel, 1992) and the learning practices that are encouraged and that shape knowledge in different classroom systems. The notion of mathematical practices attends in similar ways to the repeated actions in which people engage, but its main focus is not the learning of mathematics, but the doing of mathematics – the actions in which users of mathematics (as learners and problem solvers) engage. I have found the notion of mathematical identity – the idea that students develop relationships with their knowledge – to be extremely useful in my own analyses of learning (Boaler, 2002b). The idea of identity builds directly from studies of practice, as researchers have found that students develop identities through the practices with which they engage. Lave (1993) and Wenger (1998), for example, both claim that learning is a process of engagement with practices: Engagement in practice – in its unfolding, multidimensional complexity – is both the stage and the object, the road and the destination. What they learn is not a static subject matter but the very process of being engaged in, and participating in developing, an ongoing practice. (Wenger, 1998, p. 95)

As students engage in classroom practices, and in mathematical practices, they develop knowledge and they develop a relationship with that knowledge. Their mathematical identity includes the knowledge they possess, as well as the ways in which students hold knowledge, the ways in which they use knowledge and the accompanying mathematical beliefs and work

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practices that interact with their knowing. A focus on mathematical practices, as with classroom practices, seems generative in helping researchers understand the identities students develop, and the different ways they know and use mathematics. Authors in ESM have contributed a great deal to our understanding of the subtleties of mathematical proficiency, and the various mathematical practices employed by learners of mathematics. Special editions of the journal on proof and reasoning (see for example, ESM, 1993, 14, 4 and ESM, 2000, 44, 1–2), and constructing meaning from data (ESM, 2001, 45 (1–3)), provide good examples of such work. The notion of practices, as well as that of mathematical proficiency, may provide useful frameworks for consideration of the different aspects of mathematical work that have been delineated. The journal’s close attention to what it means to do mathematics, is in some senses not surprising, as Hans Freudenthal, one of the founding fathers of ESM, contributed a great deal to our understanding of mathematics as a problem solving act, a way of modeling and making sense of realistic situations (Van Oers, 2002). (See, for example, ESM 15, 1/2, for reflections on the ‘legacy of Hans Freudenthal’). It is critical for our field that such work continues, and that we learn more about the nuances of mathematical proficiency that include and go beyond, knowledge. Such work certainly stands as an example of theoretical development that counters the dogmatism and ideology to which S. Ball, Lave and McDermott draw our attention. For if scholars do not consider the extent and nature of mathematical proficiency, in its broadest terms, we may be reduced to the dominant ideology that pervades public rhetoric, in which mathematical proficiency is equated with the reproduction of isolated mathematical procedures. But whilst it is appropriate to commend the breadth of knowledge that has emanated from research, communicated in ESM and elsewhere, it is also sobering to reflect upon the gulf that exists between the understandings of mathematical proficiency communicated in ESM, and that which is communicated in many mathematics classrooms across the world. Greeno et al., as well as the National Academy of Education report from the US, provide some interesting proposals for ways of changing the situation and for building bridges between research and practice. These proposals include giving researchers the task of translating their findings into practical action and giving funders the task of prioritising research that takes place across academic and practitioner communities. I would add to those suggestions, a critical role for journals, such as ESM, in encouraging research that has a theory-practice dimension and giving potential authors

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new questions to consider regarding the links with, and impact upon practice, when submitting papers for publication.

5. C ONCLUSION

I was invited, in writing this paper, to reflect on the ways that ESM contributes to my work, as part of the celebration of the 50th volume of the journal. In exploring the ways that knowledge is produced and conceived – in the domains of research and mathematics – I hope to have acknowledged the critical contributions of ESM articles, in my own thinking and, more importantly, the development of our field. Articles such as those by Chevallard and Zevenbergen play an essential role in promoting theoretical awareness and reflexivity, inviting us to consider the frameworks we use and questions we raise in our research. Such articles have been extremely influential in my own work and in my teaching of future researchers, particularly masters and doctoral students. In considering the ways our field conceives of mathematical knowledge, or proficiency, I have cited just a small selection of articles that have been instructive in expanding my understanding of mathematical work. The new knowledge produced in such work feeds my ongoing research work with students and teachers in schools and my teaching and professional development work with beginning and experienced teachers. It is interesting to consider how the world would be different, how proficient students may be, if teachers, parents and policy makers, paid greater attention to the different aspects of mathematical proficiency that researchers have defined and that have been communicated in ESM. The different articles I have mentioned, along with many others, give us a glimpse of such a world, and ESM as a journal, has been instrumental in keeping such a vision alive.

N OTES 1. This is an invited paper, celebrating the occasion of ESM’s reaching its 50th volume.

R EFERENCES Artigue, M.: 1999, ‘The teaching and learning of mathematics at the university level: Crucial questions for contemporary research in education’, Notices of the AMS 46(11), 1377–1385.

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Ball, S.J.: 1995, ‘Intellectuals or technicians? The urgent role of theory in educational studies’, British Journal of Educational Studies XXXXIII (3), 255–271. Ball, D. and Bass, H.: 2000, ‘Bridging practices: Intertwining content and pedagogy in teaching and learning to teach’, in J. Boaler (ed.), Multiple Perspectives on Mathematics Teaching and Learning, Ablex Publishing, Westport, CT, pp. 83–104. Ben-Zvi, D. and Arcavi, A.: 2001, ‘Junior high school students’ construction of global views of data and data representations’, Educational Studies in Mathematics 45, 35–65. Boaler, J.: 1997, Experiencing School Mathematics: Teaching Styles, Sex and Setting, Open University Press, Buckingham. Boaler, J.: 2002a, Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and Their Impact on Student Learning (Revised and Expanded Edition ed.), Lawrence Erlbaum Association, Mahwah, NJ. Boaler, J.: 2002b, ‘The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms’, For the Learning of Mathematics 22(1), 42–47. Bourdieu, P.: 1982, ‘The school as a conservative force: Scholastic and cultural inequalities’, in E. Bredo and W. Feinberg (eds.), Knowledge and Values in Social and Educational Research, Temple University Press, Philadelphia, pp. 391–407. Bourdieu, P.: 1986, ‘The forms of capital’, in J. Richardson (ed.), Handbook of Theory and Research for the Sociology of Education, Greenwood Press, New York, pp. 241–258. Brown, M.L. (ed.): 1988: Graded Assessment in Mathematics Development Pack: Pupil Materials, Macmillan, Basingstoke, UK. Burton, L.: 1999, ‘The practices of mathematicians: What do they tell us about coming to know mathematics?’ Educational Studies in Mathematics 37, 121–143. Chevallard, Y.: 1990, ‘On mathematics education and culture: Critical afterthoughts’, Educational Studies in Mathematics 21(1), 3–28. Cobb, P., Wood, T., Yackel, E. and Perlwitz, M.: 1992, ‘A follow-up assessment of a second-grade problem-centered mathematics project’, Educational Studies in Mathematics 23, 483–504. Cooney, T.: 1999, ‘Conceptualizing teachers’s ways of knowing’, Educational Studies in Mathematics 38, 163–187. Cuoco, A., Goldenberg, E.P. and Mark, J.: 1996, ‘Habits of mind: An organizing principle for mathematics curricula’, Journal of Mathematical Behavior 15, 375–402. Dewey, J.: 1916, Democracy and Education, MacMillan, New York. Dickson, L., Brown, M.L. and Gibson, O.: 1984, Children learning Mathematics: A Teacher’s Guide to Recent Research, Holt, Rinehart and Winston, Eastbourne, UK. Dowling, P.: 1996, ‘A sociological analysis of school mathematics texts’, Educational Studies in Mathematics 31, 389–415. Ensor, P.: 2001, ‘From preservice mathematics teacher education to beginning teaching: A study in recontextualizing’, Journal for Research in Mathematic Education 32(3), 296–320. Ernest, P.: 1991, The Philosophy of Mathematics Education, Falmer, New York. Ernest, P.: 1999, ‘Forms of knowledge in mathematics and mathematics education: Philosophical and rhetorical perspectives’, Educational Studies in Mathematics 38(1-3), 67–83. Even, R. and Tirosh, D.: 1995, ‘Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject matter’, Educational Studies in Mathematics 29(1), 1–20. Greeno, J.G., McDermott, R.P., Cole, K., Engle, R., Goldman, S., Knudsen, J., Lauman, B. and Linde, C.: 1999, ‘Research, reform, and aims in education: Modes of action in

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search of each other’, in E.C. Lagemann and L.S. Shulman (eds.), Issues in Education Research: Problems and Possibilities, Jossey-Bass Publishers, San Francisco. Hart, K.M. (ed.): 1981, Children’s Understanding of Mathematics: 11–16, John Murray, London, UK. Hiebert, J.: 1986, Conceptual and Procedural Knowledge: The Case of Mathematics, Lawrence Erlbaum, New Jersey. Joseph, G.G.: 1992, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin, Harmondsworth, UK. Kieran, C., Forman, E.A. and Sfard, A.: 2001, ‘Guest editorial. Learning discourse: Sociocultural approaches to research in mathematics education’, Educational Studies in Mathematics 46(1-3), 1–12. Kilpatrick, J., Swafford, J. and Findell, B. (eds.): 2001, Adding it up: Helping Children Learn Mathematics, National Academy Press, Washington, DC. Kilpatrick, J.: 2001, ‘Understanding mathematical literacy: The contribution of research’, Educational Studies in Mathematics 47(1), 101–116. Lave, J.: 1993, ‘The practice of learning’, in S. Chaiklin and J. Lave (ed.), Understanding Practice: Perspectives on Activity and Context, Cambridge University Press, Cambridge, pp. 3–34. Lave, J. and McDermott, R.P.: 2002, ‘Estranged Learning’, Outlines 1, 19–48. Lerman, S.: 2000, ‘The social turn in mathematics education research’, in J. Boaler (ed.), Multiple Perspectives on Mathematics Teaching and Learning, Ablex Publishing, Westport, CT, pp. 19–44. Lerman, S.: 2001, ‘Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics’, Educational Studies in Mathematics 46(1-3), 87–113. Mead, G.H.: 1899, ‘The working hypothesis in social reform’, American Journal of Sociology 5, 369–371. Morgan, C.: 2000, ‘Better assessment in mathematics education? A social perspective’, in J. Boaler (ed.), Multiple Perspectives on Mathematics Teaching and Learning, Ablex Publishing, Westport, CT, pp. 225–243. National Academy of Education: 1999, Recommendations Regarding Research Priorities: An Advisory Report to the National Educational Research Policy and Priorities Board, NAE, New York. National Council for Teachers of Mathematics (NCTM): 2000, Principles and Standards for School Mathematics, NCTM, Virginia. Noss, R., Healy, L. and Hoyles, C.: 1997, ‘The construction of mathematical meanings: Connecting the visual with the symbolic’, Educational Studies in Mathematics 33, 203– 233. RAND Mathematics Study Panel: 2002, October, Mathematical Proficiency for all Students: Toward a Strategic Research and Development Program in Mathematics Education (DRU-2773-OERI), RAND Education and Science and Technology Policy Institute, Arlington, VA. Schoenfeld, A.H.: 1985, Mathematical Problem-Solving, Academic Press, New York, NY. Schoenfeld, A.: 1992, ‘Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, MacMillan, New York, pp. 334–371. Sierpinska, A. and Kilpatrick, J. (eds.).: 1998, Mathematics Education as a Research Domain: A Search for Identity. An ICMI study, Kluwer, Dordrecht, The Netherlands.

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Simon, M.: 1996, ‘Beyond inductive and deductive reasoning: The search for a sense of knowing’, Educational Studies in Mathematics 30, 197–210. Singh, S.: 1998, Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem, Anchor Books, New York. Skemp, R.: 1976, Relational understanding and instrumental understanding, Mathematics Teaching December, 65–71. Skovsmose, O.: 1994, Towards a Philosophy of Critical Mathematics Education, Kluwer Academic Publishers, Dordrecht. Thompson, A.: 1992, ‘Teachers’ beliefs and conceptions: A synthesis of the research’, in D. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, pp. 127–146. Tirosh, D.: 1999, ‘Forms of mathematical knowledge: Learning and teaching with understanding’, Educational Studies in Mathematics 38(1-3), 1–9. Valero, P.: 1999, ‘Deliberative mathematics education for social democratization in Latin America’, Zentralblatt fur Didaktik der Mathematik 98(6), 20–26. Van Oers, B.: 2001, ‘Educational forms of initiation in mathematical culture’, Educational Studies in Mathematics 46, 59–85. Vithal, R. and Skovsmose, O.: 1997, ‘The end of innocence: A critique of ethnomathematics’, Educational Studies in Mathematics 34, 131–157. Wenger, E.: 1998, Communities of Practice: Learning, Meaning and Identity, Cambridge University Press, Cambridge. Zevenbergen, R.: 1996, ‘Constructivism as a liberal bourgeois discourse’, Educational Studies in Mathematics 31, 95–113.

Mathematics Education, School of Education, 485 Lasuen Mall, Stanford University, Stanford, CA 94305-3096

STEPHEN LERMAN∗ , GUORONG XU and ANNA TSATSARONI

DEVELOPING THEORIES OF MATHEMATICS EDUCATION RESEARCH: THE ESM STORY1

ABSTRACT. In this paper we present an analysis of the articles in Educational Studies in Mathematics since 1990. It is part of a larger project looking at the production and use of theories of teaching and learning mathematics. We outline the theoretical framework of our tool of analysis and discuss briefly some of the methodological difficulties we face. We then present our findings from the analysis of the journal and we also give one example of how we ‘read’ an article, illustrating the rules whereby criteria are applied. KEY WORDS: analysis of specialised texts, educational research communities, intellectual fields of knowledge production, mathematics education research, mathematics education research publications, pedagogical models, sociology of educational knowledge, theoretical frameworks in mathematics education research

1. I NTRODUCTION : T HE POLITICS OF KNOWLEDGE PRODUCTION , PUBLICATION AND DISSEMINATION

As Educational Studies in Mathematics comes to its 50th volume one can expect that there have been many changes in the sorts of articles published within its pages through the time since Volume 1. After all, “Educational research is located in a knowledge-producing community” (Usher, 1996, p. 34). The award of research grants, the choice of candidates for doctoral supervision, the examination and award of doctorates, and of course the acceptance or rejection of articles in refereed journals such as this, are judgements made by people. Over time the values and styles and the openness, Educational Studies in Mathematics 51: 23–40, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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or otherwise, to new (or old) theories of these gate-keepers of the community change as the people move on or develop their own ideas. As mathematics education researchers (here in collaboration with a sociologist of education) we are engaged in a social science, which is always a hermeneutic activity, although some sociologists of scientific knowledge might say similar things about the natural sciences. Each editor/reviewer/examiner has her or his own interests and concerns, and her or his own trajectory of development as researcher, and these experiences are reflected in how one reads research, what one considers valid, and therefore what one allows to enter through the gates of this particular academic discipline. Of course the review of articles by three people with moderation subsequently by the editor, and the examination of PhDs by more than one examiner, will have the effect of slowing down the rate and softening the edges of change. To talk about changes over time in a field of research in terms of changes in the priorities, understandings and interpretations given by people in positions of power is, at the same time, to acknowledge a number of other aspects: the structures and social relations constituting the field as well as, perhaps, the changing strength of the boundary separating this sub-field from other research subfields within education research; changes in the relations between education research and other fields within the overall arena of research production; the wider picture of power and control relations which affect the (relative) autonomy of the intellectual field of knowledge production, establishing certain forms of social relations between, on the one hand, the official policy agencies and, on the other, agencies and agents in the field (in our case of mathematics education) of knowledge transmission, dissemination, use and reproduction (Bernstein, 1990; Morgan, Tsatsaroni and Lerman, 2002). It is also crucial to acknowledge that mathematics education research, and education research more generally, are usually located in departments of education in HE institutions, their principal purpose resulting from the training of future and in-service teachers and their overall project which historically has been a commitment to the improvement of education is affected by the overall political context (cf. Dale, 2001). This for example is evident today in the competing demands of having to act as a career academic and as a teacher educator. This picture is further complicated once we consider that educational publishing which is a crucial agency in the process of validation, authorisation and dissemination of research productions, has multiple dependencies resulting from: its symbolic control function (specialisation in discursive resources which shape consciousness); its location within the cultural field but driven/constrained by economic imperatives; and its hierarchical loc-

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ation in the division of labour – as not only diffusers but also shapers of knowledge, in the sense that they influence what counts as developments or changes within sciences (Bernstein, 1990, pp. 133–145). All the above affect the degree of authorial agency, the author as intellectual researcher and as somebody who balances the priorities/demands of the journals, the official policy or research accounting procedures, while keeping her/his credibility amongst teachers (Nixon, 1999). Furthermore, developments towards electronic publishing might further change the relations between publications, publishing and its publics. A systematic study of the changes in priorities and productions, values and styles, within a research community over time would enable the posing and examination of a whole range of questions that might otherwise be amenable only to anecdotal speculation, questions such as: – what have been the changes in the theories used by researchers in mathematics education over the years; – which theories remain in use, which have largely disappeared, and which have re-appeared; – do researchers draw more on theory than before, or less, and are the theories implicit or explicit; – what has been the ebb and flow of empirically based studies over the years; – do research theories manifest in teacher education and in school texts, and if so what transformations or recontextualisations have they gone through; and what are the principles of selection of theory(ies) in such texts and contexts? – how are new researchers ‘trained’; – what are the relations between mathematics education research and other research communities, including mathematics, science education, and educational research more generally; – what are the relations between the mathematics education research community and official decision-makers in different countries? – what are the relations between the research community(ies), its publications and the kinds of readerships constituted? Hidden behind the possibility of framing and attempting to answer these questions are some important methodological issues that would need to be addressed in such a study, for example: – what tools will be used in such an analysis; – what structure for analysis should be adopted, and what would lead to its change and development; – what are the criteria by which judgements are made?

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and so on. We have embarked on a study2 of theories, and research orientations in mathematics education the first stage of which is an analysis of a sample of papers from Proceedings of the International Group for the Psychology of Mathematics Education (PME), from Educational Studies in Mathematics (ESM) and from the Journal for Research in Mathematics Education (JRME) from 1990 to the present.3 In Morgan et al. (2002) we discussed two issues regarding the field of mathematics education research: its status within educational studies and hence within the general field of intellectual production; and the way its knowledge is organised, its internal structure (p. 450). Our study is mainly focused on the latter, whilst recognising that the former is expected to be always inscribed in the latter. As it develops, we will be particularly concerned to examine how the shifts between different models of pedagogy occur as changes in the orientation, structure and boundaries of the field of mathematics education knowledge production changes (p. 451). This will be one of the results of the whole study. At this stage we can only make general comments about our findings. Thus the main focus of this article will be a report on our analysis of the output from ESM in this period. We will be attempting here to answer the following main question: How is the field of mathematics education research shaping up, as represented in the journal? This overall question will be approached in terms of the following sub-questions: 1. What are the topics, issues, priorities and emphases addressed by authors, and what changes have there been on these over time? 2. Who are the addressors and the addressees of the journal articles? What changes have there been in the perceived addressees over time? 3. What are the relations of mathematics education research with other fields in educational and social inquiry? 4. What are the field’s relations to official agencies and how has this changed over time?

2. M ETHODOLOGICAL ISSUES

In this article we will present only a brief sketch of methodological aspects of the project (for further elaboration on methodology see Tsatsaroni, Lerman and Xu, 2003). Whilst the choice of years of publications to analyse is to some extent arbitrary, it is based on two factors: we wanted to bring the analysis up to the present day; and we are most interested in the years since the entry of more social theories into the field (see Lerman, 2000). Given the size

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of the task, even just analysing the two journals ESM and JRME and the Proceedings of PME over 12 years it was necessary to make a systematic selection. In relation to ESM we have chosen to examine every article from every second ‘book’ (using the term ‘volume’ or ‘issue’ risks confusion). We consider that, in terms of randomness, the articles published bear no relation to each other or to authors’ names, except for the Special Issues. In these cases, we conjecture that the only category affected would be the topic since methodologies, theories and all the other elements of our analysis will vary across these articles as much as in any other book. We have also examined the Special Issues separately for topic. Were there time, we would have liked to look back across the whole 50 volumes of ESM and we expect that the changes would be quite substantial. We can expect less changes over a 12 year period, but our project’s focus is the field of mathematics education research more generally than just that represented by one journal. At a later stage the project will also involve examining a selection of teacher education texts over the same period in order to be able to comment upon the process of recontextualisation taking place in the community. To this stage we have developed a tool4 of recording and analysing the specialised texts of the research community. We have not used any existing methods of textual analysis but we have tried to construct a tool ourselves, drawing broadly on Basil Bernstein’s work and using in particular his latest work on intellectual fields and knowledge structures (Bernstein, 1999, 2000). For example, we have a view about the field as a series of positions, as a horizontal knowledge structure. We would therefore view new theories as, in general, positioned alongside other theories and not replacing them, as we might expect in the development of theories in science. This tool has changed, and continues to change, as we interrogate more articles and find our categories inadequate or requiring modification. A key factor has been the development of justifications for judgements, what Bernstein (2000) calls recognition and realisation rules, for what makes us place an aspect of an article in one category or another in an explicit manner. We are concerned that this project should be an empirical, descriptive study and at the same time to generate a language capable of showing the effects of that which it describes. Certain structural features, drawn from a variety of places, have been used to construct the tool. Firstly, we have made a distinction between an orientation towards the theoretical or towards the empirical, according to which domain has been privileged. Articles in the first category may move to the empirical to illustrate the theory, but in this category the intention of the article is to present and perhaps to develop theory. Similarly, articles

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that are orientated towards the empirical may well draw on theory, but their orientation is towards describing and perhaps informing school practice, policy, or other site of practice. In the first category we will then analyse how theory is used, whether it is supported or modified, and whether theories from other fields are used. In the second category we will look at what is the focus of the empirical, whether school practice, researchers’ practice, etc., and what methodology, data collection and analysis have been used. We also look at the relationship between theory, where the authors have drawn on theory either explicitly or implicitly, and the empirical, in the sense of whether the theory informs the empirical, is informed by the empirical, or there is a dialectic between them. Subsequently, we explore the question of whether the author(s) overtly adopt a particular position, such as feminist, post-modern, or other, and we identify the addressees of the article. We then examine the pedagogical model projected/promoted in the paper of the authors, where one has been identified. Here there are three subquestions derived from Morgan et al. (2002). The first is whether the orientation is towards a knowledge mode or towards pupils, the knower mode (see also Lerman and Tsatsaroni, 1998 and Maton, 2000). The second sub-question concerns the strategy (cf. Dowling, 1998; cf. Brown, 1999), and whether the authors look towards what is present or absent in students’ texts and whether they make localised to specialised comments. The third sub-question concerns the nature of the boundary between the everyday and specialised mathematics discourse and whether the boundary is presented as strong or weak. Finally, we ask three questions in order to locate the article within the field: what is the position within the official education discourse, i.e. towards the policy-makers; what is the position vis à vis the mathematics education research field; what is the position vis à vis other fields of knowledge production? We anticipate further restructuring of the analysis as the research proceeds. Nevertheless, we feel that the work we have done enables us to make some interesting comments in relation to how the field is shaping up through looking at the articles in this journal. One key reservation must be made, however. Researchers, in choosing a forum for the dissemination of their research, make assumptions about the type of audience constituted by the participants at a conference or the readers of a particular journal, and their particular interests and orientations (Burton and Morgan, 2000). They are likely to orientate their style accordingly. From the analysis of one journal, here of ESM, we cannot comment in an informed way on how researchers do change their writing for different perceived audiences; we

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may be able to do so after the analysis of all three fora, ESM, PME and JRME, is completed.

3. ESM SPECIAL ISSUES

The topics of the Special Issues since 1990 are listed in Table I. The journal does not solicit Special Issues from the community, other than a general invitation for proposals. In the words of the former Editor-in-Chief: My experience was that once such a proposal had been made (by a member of the community), it generally got accepted after some discussion and revision. . . The high success rate of proposals reflects the fact that they generally came from very experienced people in the field, often drawing on special conferences or working groups which had been taking place, and therefore focusing on themes where there were likely to be interesting new things to say. It also reflects an attitude amongst the ESM editors which I would describe as one of openness to different approaches, as long as there is some clear sense of intellectual discipline behind them. (Ruthven, 2002, personal communication, our addition in italics)

In this sense, although ESM and other journals (as well as conference papers, research funding, etc.) clearly play a crucial role in the development of the field it does not drive what appears, either in the journal in general or in the Special Issues. Nevertheless, the comments above regarding the proposers indicate a tendency to perpetuation of mainstream themes and issues, whilst also facilitating the wide dissemination of work in those domains. A significant event in this period has been the formalising of the relationship between PME and the journal through the publishing of Special Issues, beginning with the one edited by Jones, Guttierez and Mariotti, (2000) that are products of Working Groups at conferences and in one case a development of a plenary panel, edited by Kieran, Forman, and Sfard (2001). A link such as this between a conference and a journal marks an important moment in the development of the field that will need to be evaluated at a later stage in our project. What can we say about the field from the topics of the Special Issues? Most present examples of substantial bodies of work within the community. Some of them have a specific purpose, to bring new ideas into the field. We can note in particular those edited by Lerman (1996), Boero (1999) and Kieran, Forman, and Sfard (2001). We can ask of these particular volumes when they are bringing a new approach to the practice of mathematics teaching and when to research. When talking about research we can ask whether the focus is on theory or on other aspects of research.

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TABLE I Special Issues since 1990 Year

Vol.

No.

Topic and editors

No. of papers

23 23

1 5

4 6

24 24

1 4

25

1–2

1994

26

2–3

1995

27 28 29

4 3 2

1996

31

1–2

1997

33

2

Classroom Dynamics (Colette Laborde) Constructivist Teaching: Methods and Results (Ernst von Glaserfeld) Design of Teaching (Alan Bell) Aspects on Proof (Gila Hanna and Niels Jahnke) The legacy of Hans Freudenthal (Leen Streefland) Learning Mathematics: Constructivist and Interactionist Theories of Mathematical Development (Paul Cobb) Assessing Mathematics (Leone Burton) Mathematics and Gender (Gilah Leder) Advanced Mathematical Thinking (Tommy Dreyfus) Socio-Cultural Approaches to Mathematics Teaching and Learning (Stephen Lerman) Computational Environments in Mathematics Education (Richard Noss)

1990 1991 1992

1993

1998 1998/99 1999 38

1–3

39

1–3

2000

44

1–2

2001

45

1–3

46

1–3

48

2–3

Forms of Mathematical Knowledge: Learning and Teaching with Understanding (Dina Tirosh) Teaching and Learning Mathematics in Context (Paolo Boero) Proof in Dynamic Geometry Environments (Keith Jones, Angel Gutierrez, Maria Alessandra Mariotti) Constructing Meanings from Data (Janet Ainley and Dave Pratt) Bridging the Individual and the Social: Discursive Approaches to Research in Mathematics Education (Carolyn Kieran, Ellice Forman and Anna Sfard) Infinity – The Never-ending Struggle (David Tall and Dina Tirosh)

5 6 8 6

6 7 5 8 4

10

12 6

7 7

8

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We can ask further questions of those that refer to practice, and indeed of all the Special Issue volumes, whether that orientation is to a topic in mathematics, to pupils, or to teaching. It is not always easy to classify uniquely by these questions as the volumes may address a number of these orientations within the range of papers included in the volume. We can see a strong focus on content across the years, from Aspects of Proof in 1993 to Infinity in 2001, as well as a focus on pupils’ learning and on teaching. In terms of our sub-questions, we can notice that there is an absence of an engagement of the mathematics education research community with other agencies, in terms of policy or politics. There is no discussion of curriculum, and there is one issue on assessment. Mathematical pedagogy appears to be the main interest of the research community, as evidenced in the Special Issues. We will want to comment on the relations between the community and official agencies but we will do so below. As a final comment, we note the recent increase in the number of Special Issues and in the number of articles in them. The PME books play a major part in this, but what it indicates about the state of the field is of interest. For example, some commentators have called for more unification of the work in the community and these volumes may play a part in this. Whilst reflecting on-going research, by drawing together research perspectives and methods and by posing further research questions these volumes also influence future research. On the other hand, it might indicate an increased pressure on researchers to move on from presenting a paper at a conference to seeing it published in a (refereed) journal.

4. A NALYSIS OF ARTICLES

As researchers we are particularly interested in and challenged by the methodological problems in this research. In our description of our findings we struggle with the criteria by which we categorise aspects of the articles, as we mentioned above. There is no space here fully to describe the rules we have developed for each category. We will, however, give just one example at the end of the overview to demonstrate our methodology. We will also not describe all the categories that our research tool investigates, for lack of space. We will focus on areas we think of most interest to readers. 4.1. Theories Our analysis shows that just over two thirds of all articles in ESM have an orientation towards the empirical, with a further 9% moving from the

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theoretical to the empirical, and this has changed little over the years. A little less than three quarters are explicit about the theories they are using in the research reported in the article. Again this has not varied across the years. The theories that are used have changed. We can notice the following: – The range of theories or approaches used has developed over the years. Discourse analysis, hermeneutics, philosophy of mathematics, psychoanalytical theories and the sociology of Bourdieu and of Bernstein have all appeared in the last 6 years of our analysis. We will revisit the issues of theories below, when we consider the relations between the field of mathematics education research and other intellectual fields. – Below we will make a distinction between psychology and mathematics, on the one hand, as theories traditionally drawn upon by authors, and social theories on the other (see Lerman, 2000). Whilst social theories have been used (including social constructivism, Vygotskian theories, symbolic interactionism and others) throughout the period, we now see a much higher percentage drawing on these theories, from 9% average from 1990 to 1995, to 34% from 1996 to 2001. – As is perhaps appropriate given the anniversary being marked in this issue, Freudenthal’s ideas have received an increased interest in the last 3 years. In our analysis of how authors have used theories we have looked at whether, after the research, they have revisited the theory and modified it, expressed dissatisfaction with the theory, or expressed support for the theory as it stands. Alternatively, authors may not revisit the theory at all, content to apply it in their study. We have found that more than three-quarters fall into this last category, just over 14% revisit and support the theory, whilst less than one percent propose modifications. No authors in our sample ended by opposing theory. This pattern has not changed over the years. 4.2. Topics We have analysed the topics addressed in three ways: by sector of education (primary, secondary, etc.); by mathematics topic; and by whether its focus is towards teaching, that is, content, or learning, including its acquisition and assessment. Secondary/high school age is the predominant interest of authors published in ESM, with 44% of all articles in the sample, primary/ elementary being the topic for 37%, with a further 9% addressed the overlap of primary/ elementary and secondary/high school age. Students at Univer-

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TABLE II Analysis of mathematical topics Topic Problem solving Number concept Arithmetic computation Estimation Fraction Ratio and Proportion Probability Algebra Geometry Proof Function and graph Advanced maths Statistics

Number

Percentage

21 8 6 3 5 5 5 8 12 2 8 7 5

19 7 5 3 5 5 5 7 11 2 7 6 5

sity are the focus of 8% of papers, with only a slight tendency towards an increase in this focus in recent years. Otherwise the pattern is consistent across the period. The spread of mathematical topics is very wide, as may be expected. The topics, totals and percentages appear in Table II. Of interest is the evidence that problem solving has the largest focus, followed by geometry. Most of the articles on problem solving appeared between 1994 and 1998, during which there were no special issues on this. With regard to focus we used the following categories: researchers’ practice; student teachers’ practice; policy; and school practice, with a further category for other focuses if they appeared. School practice is further split between agent/person, as teacher or pupil, and content, with content being sub-divided again into Bernstein’s three message systems, termed transmission5 , acquisition6 , and evaluation. There are very few articles in our sample of ESM from 1990 focusing on researchers’ practice although there is a small increase in frequency since 1996. Similarly there have been very few articles focusing on student teachers’ practice and an even smaller number on policy. The main two categories have been pupils as agents, an average of 34% of articles, and acquisition, an average of 34%. Teachers as agents had an average of 14% and transmission just 6%. Apart from the Special Issue on assessment, only two articles appeared in our

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sample focusing on evaluation. The emphasis on pupils from a content point of view has increased in the last 6 years but emphasis on acquisition has remained constant. 4.3. Addressees An analysis of addressees is not an easy matter and requires careful consideration of criteria for classification. At this stage, we have used the authors’ own references, usually in the discussion or conclusions section, regarding to whom the research speaks. We have drawn on four categories here that have arisen purely pragmatically: researchers; researchers and teachers; researchers and teacher educators; and researchers and policy-makers. Almost three-quarters are addressed to researchers and teachers, almost a fifth to researchers alone, the remainder being almost all to researchers and teacher educators. The two categories, of researchers and teachers and of researchers, appear more frequently in the last 3 years. It is interesting to speculate why it is that most are addressed to researchers and teachers, given that it is probably the case that few teachers, other than those undertaking higher degrees or carrying out research, read the journal. It is also interesting to note how few articles are addressed to policy-makers. 4.4. Relations with other fields In this question, which relates to issue 1, theories, discussed above, we have examined what resources researchers draw upon in their research as it appears in publication. We emphasise the latter because it is not uncommon to find a substantial and informed review of literature in an article, in which the range of theoretical resources drawn on by others are noted, but then for the authors not to use any theory themselves, at least explicitly. One might suggest that an element of accepted style for publication may account for this phenomenon. As we discussed above, research in mathematics education is a horizontal knowledge structure, we might also say it is a region (Bernstein, 2000) in that it is a field that looks inwards to theoretical fields (psychology, sociology, anthropology etc.) and outwards towards practice. New theories and perspectives often develop from drawing upon different intellectual fields from those used before, or alternative aspects of those fields. Thus, they do not replace former theories but sit alongside them, what has been described elsewhere as: “the addition of a new language, an additional segment, rather than greater generality and integrative potential” (Morgan et al., 2002, p. 450). We might suggest that there is a connection here with creating identities, making a unique space from which to speak in novel ways, but we would need another study to substantiate and instantiate this claim. The relationships between mathematics education research

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35

and other fields are complex but it is vital to attempt to identify these fields (see e.g. Higginson, 1980 for an attempt to describe how the fields relate to each other and to mathematics education research). Following Kilpatrick (1992) we will take psychology and mathematics as having a long history as intellectual fields at the heart of the discipline and map other intellectual resources as they have appeared in our sampling. We thus listed the theoretical fields drawn upon by authors, based on their explicit references in Table III. We can say that there has been a substantial increase in the number of fields in 1998, 1999 and 2000, it is too early to say whether this trend will continue, as 2001 shows a slight dropping off. As we have mentioned before, it appears that quite a range of social and sociological theories are being used increasingly by authors. 4.5. Relations with agencies As we mentioned above, there are very few articles in our sample that have addressed policy issues and addressed them to policymakers as well as the research community, one in each of 1994, 1998, and two in 2001. The relations between policymakers, or the Official Pedagogic Recontextualising Field (OPRF) as Bernstein (2000) calls it and the mathematics education research community and its activity within the intellectual field of knowledge production and within the Unofficial Pedagogic Recontextualising Field (UPRF) varies substantially across the world. In some cases, as presently in the UK, the UPRF appears to have very little or no influence on the OPRF, in that they appear to make their own selection of research on which to draw for their policies. We conjecture that the international nature of ESM leads to few articles being offered or accepted that are location-specific. We have found the same phenomenon in our analysis of PME proceedings. We conjecture, further, that although JRME is also an international journal it tends to be US-focused, so that authors generally see themselves as addressing one community and will therefore not shy away from attempting to engage with the OPRF in the pages of the journal. We have yet to carry out that stage of our project.

5. M ETHODOLOGY – RECOGNITION AND REALISATION

We will take, as an example, one article at random and indicate the rules by which we allocate classificatory criteria. We invite readers to follow and critique those rules. The article was published in volume 28 and the authors are Ma Tzu-Long Yang and Paul Cobb (1995). Looking first at theory, the authors are explicit about their use of Vygotskian theory (p.

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TABLE III Theoretical fields Year

Theoretical fields other than educational psychology and/or mathematics

1990 1991 1992

Attribution theory, science education

1993 1994

1995

1996

1997

1998

1999

2000

2001

Symbolic interaction Ethnomethodology Didactical situations Vygotsky Cognition in self-referencing systems Poststructuralism Hermeneutic phenomenology Vygotsky Feminist theory Transactional theory of stress Historical research Vygotsky Sociology (Bourdieu) Philosophy of mathematics Ethics Vygotsky Metaphor Transitional phenomena (Winnicott) Sociology (Bernstein) Situated cognition Gestalt Vygotsky Philosophy of mathematics Cognitive science Sociology (Bernstein) Discourse theory Psychoanalysis Vygotsky Embodied cognition Sociology (Ricoeur) Educational theory – Grounded Theory Situated cognition Vygotsky Process of psychological curtailment Educational theory – Collaborative Action Research Social psychology Discourse theory Ethnomethodology Situated cognition Vygotsky Motivation theory Embodied cognition

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4) and their finding inspiration in Bishop’s work on enculturation (p. 3) in particular. It is a paper orientated to the empirical in that it looks to a study of Taiwanese and USA children to create theories for differences in competence in arithmetical thinking (p. 4). Its focus is on school practice and, further, looks at pupils’ cognition. Theory informs the empirical as the researchers develop sociocultural explanations for the differences the two studies reveal (pp. 27–30). The authors address teachers, in providing implications for classroom practice (p. 31) but also researchers in their emphasis on what is revealed by their theoretical orientation (p. 29). Finally, the authors draw on an intellectual resource beyond mathematics education research as we defined it, that is, beyond mathematics and psychology, Vygotskian theory, and they do not seek to modify or critique it. They do not engage with official fields. We categorise the location of their research as using theory. 6. S UMMARY

Finally, then, what can we say about “How is the field of mathematics education research shaping up, as represented in the journal”? We might suggest that the field exhibits a weak grammar, in that we can see a proliferation of new specialised languages, creating new positions within the field. As seen in ESM, the field of UPRF is active and developing, but it does not engage with the OPRF; we could say that, as represented in ESM over these years research sees itself as agent committed to UPRF development but not to OPRF. Researchers use a growing range of intellectual discourses as resources, are generally explicit about the theory they use, and they generally do not critique those theories. The focus is, in the main, towards learners, either from a curriculum point of view or a pedagogic point of view, and authors address the research community and teachers, although we have no explanation for the latter, other than, as argued earlier, in terms of the location, the purpose and the context in which mathematics education research is being carried out. An orientation towards social theories of one kind or another is increasing. As many of these languages or theories are drawn from the field of sociological inquiry, broadly speaking, one can hypothesise that the new positions created in the field will be supporting more radical pedagogical forms; they will be orientated towards developing more democratic and participatory forms of teaching, and will be concerned with empowering – i.e., giving ‘voice’ to groups of students (or teachers) thought to be silenced within the official pedagogic discourse – rather than focusing on the elaboration of their object of study (Bernstein, 1990; Maton, 2000). Although much more work is needed to support this

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conjecture, perhaps, we can also venture the hypothesis that this might be an additional explanation of why this research community – as presented in the journal – has seen its task as one of supporting and guiding teachers rather than, for example, the policy makers. Furthermore, the journal seems to be explicitly reactive to the interests and concerns of researchers in the mathematics education community and does not seem to direct changes and developments. The impression is of a journal which aims to relay rather than construct ‘voices’. By allowing different positions from within the field of mathematics education research to appear, as published work, it constitutes a space where authorial voices can be re-presented. However, it might be argued that this space of freedom, which constitutes researchers as authors and readers and teachers as readers, might serve as well, on the one hand to reproduce the hierarchies between what Bernstein (1990) calls producers and reproducers of discourse and, on the other hand, to establish and legitimate itself as a mainstream academic journal. Indeed, the absence of active encouragement or initiative towards opening up debates might support such a reading. What other characteristics of the research culture get to be promoted (eg. the number of writers authoring a paper/changes within this period/emerging culture of research teams) given the current political context in which a distinction and hierarchy is developing between research and non-research active academic identities, is hard to say (cf. Hey, 2001) since we can obtain only limited information from a textual analysis alone. Without information as to what is rejected, and with what criteria, we cannot fully appreciate what is really produced. Finally, while acknowledging that the changes in reviewers over the years will have had its, perhaps, immeasurable effects on how the journal, and the field, appear in this 50th volume, we should also give thought to the claim that “. . . most academics remain deathly silent about the conditions of their own production” (P. Rabinow, quoted in Hey, 2001, p. 27) and to the claim that “[u]ntil we can bring to the surface and publicly discuss the conditions under which people are hired, given tenure, published, awarded grants and feted, ‘real’ reflexivity will remain a dream” (R. Gill, quoted in Hey, 2001, p. 67). We hope that the study reported here can help towards achieving greater reflexivity.

N OTES 1. This is an invited paper, celebrating the occasion of ESM’s reaching its 50th volume. 2. The project, which commenced in October 2001, is supported by the Economic and Social Research Council in the UK, # R000 22 3610. Its full title is: “The production of

DEVELOPING THEORIES OF MATHEMATICS EDUCATION RESEARCH

3. 4. 5.

6.

39

theories of teaching and learning mathematics and their recontextualisation in teacher education and education research training”. For other kinds of data to be collected, such as data from interviews with editors of Journals see reference in Note 2, above. See http://www.sbu.ac.uk/cme/Paperforpeercomment.doc Note the term is intended to be general and does not distinguish between forms of pedagogy. ‘Transmission’ teaching is often taken as conveying a pejorative meaning when compared to pupil-centred styles: we are using ‘transmission’ in a more general sense to represent a focus on what the teacher is doing, not to engage in any debate on preferred styles. Similarly, acquisition is to be taken as a general term for the range of styles of learning.

R EFERENCES Bernstein, B.: 1990, Class, Codes and Control, Vol. 4, The Structuring of Pedagogic Discourse, Routledge, London. Bernstein, B.: 1999, ‘Vertical and Horizontal Discourse: an Essay’, British Journal of Sociology of Education 20(2), 157–173. Bernstein, B.: 2000, Pedagogy, Symbolic Control and Identity: Theory, Research, Critique, Rowman and Littlefield, Maryland. Brown, A.J.: 1999, ‘Parental participation, positioning and pedagogy: a sociological study of the IMPACT primary school mathematics project,’ Collected Original Resources in Education 24(3), (7/A02-11/C09). Burton, L. and Morgan, C.: 2000, ‘Mathematicians writing’, Journal for Research in Mathematics Education 31(4), 429–453. Dale, R.; 2001, ‘Shaping the sociology of education over half-a-century’, in J. Demaine (ed.), Sociology of Education Today, Palgrave, Basingstoke. Dowling, P.: 1998, The Sociology of Mathematics Education, Falmer, London. Hey, V.: 2001, ‘The construction of academic time: sub/contracting academic labour in research’, Journal Education Policy 16(1), 67–84. Higginson, W.: 1980, ‘On the foundations of mathematics education’, For the Learning of Mathematics 1(2), 3–7. Kilpatrick, J.: 1992, ‘A history of research in mathematics education’, in D.A. Grouws (ed.) Handbook of Research on Mathematics Teaching and Learning, MacMillan, New York, pp. 3–38. Lerman, S.: 2000, ‘The social turn in mathematics education research’, in J. Boaler (ed.) Multiple Perspectives on Mathematics Teaching and Learning, Ablex, Westport, CT, pp. 19–44. Lerman, S. and Tsatsaroni, A.: 1998, ‘Why children fail and what mathematics education studies can do about it: The role of sociology’, in P. Gates (ed.) Proceedings of the First International Conference on Mathematics, Education and Society (MEAS1), Centre for the Study of Mathematics Education, University of Nottingham, pp. 26–33. Maton, K.: 2000, ‘Languages of legitimation: the structuring significance for intellectual fields of strategic knowledge claims’, British Journal of Sociology of Education 21(2), 147–167.

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Morgan, C., Tsatsaroni, A. and Lerman, S.: 2002, Mathematics teachers’ positions and practices in discourses of assessment, British Journal of Sociology of Education 23(3), 443–459. Nixon, J.: 1999, ‘Teachers, writers, professionals. Is there anybody out there?’ British Journal of Sociology of Education 20(2), 207–221. Tsatsaroni, A., Lerman, S. and Xu, G.: 2003 A sociological description of changes in the intellectual field of mathematics education research: Implications for the identities of academics. Paper presented at American Educational Research Association, Chicago, April. Usher, D.: 1996, ‘Textuality and reflexivity’, in D. Scott and R. Usher, (eds.), Understanding Educational Research, Routledge, London, pp. 33–51. Yang, M.T.-L. and Cobb, P.: 1995, ‘A cross-cultural investigation into the development of place-value concepts of children in Taiwan and the United States’, Educational Studies in Mathematics 28(1), 1–33.

Faculty of Humanities and Social Science, Division of Education, 103 Borough Road, London SE1 0AA, UK, ∗

Author for correspondence: E-mail: [email protected]

DEREK FOXMAN1 and MEINDERT BEISHUIZEN2

MENTAL CALCULATION METHODS USED BY 11-YEAR-OLDS IN DIFFERENT ATTAINMENT BANDS: A REANALYSIS OF DATA FROM THE 1987 APU SURVEY IN THE UK

ABSTRACT. This paper describes the reanalysis of data obtained in 1987 on the mental calculation strategies used by a sample of 11-year-olds in the course of a national survey of schools in England, Wales and Northern Ireland. The mental skills test was administered to pupils in a one-to-one situation and the reanalysis made use of classifications of mental strategies developed in the past decade in international research. These pupils were a subsample of the main sample of about 10 000 who took a written test of mathematics in the survey, 247 taking both tests. The scores on the written test were used to distribute these pupils into three bands of attainment in order to compare the frequency and effectiveness of the strategies used by pupils of different levels of attainment. Eight of the reanalysed questions are discussed, each of them involving one of the four basic arithmetic operations. Some are purely numerical and in others the numbers are in a context. Top band pupils mostly preferred sequential strategies which leave one of the numbers in the calculation complete, while the Bottom band generally preferred to split complete numbers and operate separately on the components of the numbers thus partitioned. The most popular strategy for Middle band pupils in several questions was the standard algorithm used mentally. Complete number strategies were the most successful for pupils in all three bands. Possible implications for the National Numeracy Strategy in Britain are discussed. KEY WORDS: addition, attainment bands, children’s mental strategies, context, division, mental calculation, multiplication, subtraction

I NTRODUCTION : T HE ORIGIN OF THE REANALYSED DATA According to Anita Straker (1999, p. 43), “The ability to calculate mentally lies at the heart of numeracy”. This tenet has been written into the National Numeracy Strategy currently being implemented in primary schools in Britain and now progressing into secondary schools. The importance of mental calculation in the development of early numeracy has long been a tradition in several continental countries, but in Britain, some 20 years ago, the government appointed committee of inquiry into the teaching of mathematics in schools (often known as the Cockcroft Committee after its chairman Dr W.H. Cockcroft) stated in its report (DES, 1982) that mental calculation had been neglected for some years in classrooms and should return. However, the results of national and international surveys in the dozen Educational Studies in Mathematics 51: 41–69, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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or so years following the publication of the Committee’s report suggested that schools were not focusing on calculation skills. International surveys of the 1980s and 1990s continued to show that number was a weak area in England relative to several continental countries (e.g. Foxman, 1992). Furthermore, national surveys of the mathematics performance of 11- and 15-year-olds, carried out in 1982, the year the Cockcroft Report was published, and again in 1987, found that scores in calculation on written tests declined in that interval. These national mathematics surveys were carried out by the National Foundation for Educational Research (NFER) on behalf of the Assessment of Performance Unit (APU) a Unit at the then Department of Education and Science (DES). The Unit was established in the mid-1970s and commissioned surveys in several core subject areas of the curriculum from independent research agencies. It was terminated when the National Curriculum and its associated programme of testing were introduced in 1988. The APU mathematics team developed a variety of assessments; accounts of the surveys and tests used can be found in the APU reports (see, for example Foxman et al., 1985, 1991). The 1987 APU surveys included tests of mental calculation skills administered in a one-to-one situation by experienced teachers who were therefore able to ask the children about the methods they had used to carry out calculations and record them. The evidence of a decline in the calculation scores of both 11- and 15-year-olds between 1982 and 1987 suggests that the mental calculation methods used by pupils in the 1987 surveys were untaught, and probably unknown to many teachers. Some of the results of the APU mental skills tests of both age groups were included in the final APU report (Foxman et al., 1991), but many of the responses to questions were left unclassified. We thought that some of these might now fit into the classificatory schemes developed by researchers in the 1990s decade (e.g. Beishuizen et al, 1997; Fuson et al., 1997; Thompson, 2000a). The NFER archives yielded the original interview data on the 256 age 11 pupils who took the mental skills test, all but nine of whom had also taken a written test of concepts and skills. This article is based on the data from the 247 11-year-olds who took both a written test of concepts and skills and the mental skills test. The article has four main purposes; to explore further the data from the 1987 APU survey in relation to:

• the mental methods used by 11-year-olds at a time when the methods were almost certainly untaught, and relate them to categories of response developed in the past decade;

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• the methods used by pupils in different attainment bands, with the bands established by their scores in the written tests of concepts and skills, a measure independent of the mental test score; • a comparison with Dutch research findings on the differential effectiveness of mental calculation methods; • implications for the teaching of mental strategies.

1. T HE ADMINISTRATION OF THE MENTAL SKILLS QUESTIONS IN THE

1987 APU SURVEYS All the APU mathematics surveys included a number of practical tests that were administered orally by about 30 experienced teacher-assessors in one-to-one sessions with pupils. These teachers were nominated by the Local Education Authorities in which their schools were situated and trained to administer the tests by the NFER research team at a residential briefing conference lasting two and a half days. All the practical tests were untimed so that pupils were given as much encouragement as possible to demonstrate what they knew and could do. The one-to-one situation with its oral delivery of questions provided opportunities for controlled interaction between assessor and pupil. This interaction, especially the ‘probe’ question “How did you get your answer”, provided rich information on the methods pupils used to tackle the questions asked. The sample of 247 11-year-olds is likely to have been reasonably representative of the populations included in the surveys. However, the practical test subsamples were not drawn from the main sample on a totally random basis for practical reasons: very small schools with fewer than six pupils in the age range and schools situated in relatively remote areas were not included. The mean score and standard deviation of the subsample were close to those of the main sample of about 10 000 pupils, but there was an imbalance in the gender representation: there were 133 boys and only 114 girls. Assessors were provided with a ‘script’ for each topic that included instructions about what to say to the pupil and how to present materials and questions. The questions were printed in a booklet that contained one question per page, and an interviewee was asked to read out each question before being asked to do the calculation. Pupils were told by the assessor: I’m going to give you some questions that I want you to work out in your head. The questions are in this booklet. After each one I’m going to ask you how you worked out your answer – that’s not because you have got it wrong (or got it right), I’m just interested in how you worked things out. Here is the first question. Please read it to me and then work out the answer.

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When pupils had given an answer they were asked how they had worked it out. Assessors recorded the responses in as much detail as possible. Sometimes pupils who had given an incorrect answer realised their error when describing their calculation; in such cases it was the second answer that was counted in the tally of correct responses. The original analysis recorded in the final report of the APU mathematics team (Foxman et al., 1991) left about 50 per cent of responses unclassified; the present reanalysis reduced this proportion to about 15 per cent. 1.1. The formation of the three attainment bands In the analyses described below pupils were assigned to attainment bands according to their scores on written tests of concepts and skills. In the 1987 survey all the 11-year-olds in the main sample of about 10 000 pupils took one of 20 different written tests of concepts and skills. Each of the concepts and skills tests comprised about 50 short response items from 3 of the 12 subcategories of content. (The 12 subcategories were organised in five main categories: Number, Measures, Geometry, Algebra, Probability and statistics. The total survey pool consisted of about 600 items.) In order to take account of their differing difficulties the tests were linked so that test scores could be placed on a common scale with a mean of 100 and standard deviation of 15. (See Appendix 2.1 of Foxman et al., 1991 for details of the survey test design). The written test scaled scores of the 247 pupils taking the mental skills test ranged from 68 to 128. Mental Skills test scores (out of 10) of sample pupils correlated 0.67 with their written test scaled scores. A third of the Total sample was about 82 pupils. Each Third had approximately that number of pupils but the boundary between Thirds was arranged to be at a point of change in the written test scaled score as shown in Table I. TABLE I Written test scaled score ranges for each third of the sample Third Top Third Middle Third Bottom Third

Written test scaled score 107–128 95–106 68–94

n 81 80 86 247

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2. T HE CLASSIFICATION OF MENTAL STRATEGIES IN THE REANALYSIS

In the 1980s, research on mental calculation strategies focused first on addition and subtraction with numbers under 20 (Carpenter, 1985). The 1990s saw an extension to strategies with larger (two-digit) numbers up to 100, research that was useful for the reanalysis of the APU data. For instance, Beishuizen et al. (1997) and Fuson et al. (1997) published rather similar classification schemes, based on research respectively in Holland and the USA, which we will use as a framework (Table II). We have added Thompson’s (2000a) strategy labels used in a recent publication for British teachers. Beishuizen’s strategy labels (Table II) are short acronyms introduced for ease of coding. The sequential strategy of Jumping in tens in either direction from any point in the number sequence is called N10 (meaning: N=Number; N+10 or N–10). Splitting off tens and units according to their place value and adding or subtracting them separately is called 1010 (meaning: 10+10 or 10–10). These are the two main mental calculation strategies for larger numbers, as the aforementioned research pointed out. We will use Fuson’s et al. (1997) full names Sequence-tens for N10 and Separate-tens for 1010, as they express concisely and precisely the fundamental difference in dealing with the tens (or hundreds) in larger numbers. Variants illustrated in Table II are N10C using Compensation, and 10S which begins with the tens like the split method 1010, and then handles the units sequentially. The latter strategy 10S is also called a ‘mixed method’ (Fuson et al., 1997; Thompson, 2000a). The strategy 10S is a special case as it could be considered as a transition between 1010 and N10 (Beishuizen and Anghileri, 1998; Beishuizen, 2001), when carried out as a correct adaptation of the 1010 impasse in the subtraction 4–6 (cf. Table II). However, it is often not the result of such an adaptation, but due to a misunderstanding or confusion when both units are subtracted (invalid: –4, –6; cf. Table II; cf. Fuson et al., 1997, p 152). A last type is A10 (Adding-on to a nearest ten) which is used a lot in the strategy Complementary Addition, and is quite popular in the UK as an alternative to Subtraction. Dutch research has given a good deal of attention to mental calculation strategies under the influence of the ‘realistic mathematics education’ movement. This emphasized that mathematics learning should always start from informal strategies which are promoted both by realistic context problems and by mental calculation (Gravemeijer, 1997; Treffers and Beishuizen, 1999). A similar philosophy can be found in the schools

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TABLE II Mental addition (38+26) and subtraction (64–26) strategies up to 100 Beishuizen

Thompson

Examples of strategies

N10

Sequencing (Jump method)

38+20 → 58; +6 → 64 64–20 → 44; –6 → 38

N10C

Compensation

38+30 → 68; –4 → 64 64–30 → 34; +4 → 38

A10

Complementary addition

38+2 → 40; +20 → 60; + 4 → 64 26+4 → 30; + 30 → 60; + 4 → 64 (answer: 4 + 30 + 4 → 38)

10S

Mixed method

30+20 →50; +8 →58; +6→ 64 60–20 →40; +4 →44; –6 →38 (invalid: 40–4 → 36–6 → 30)

1010

Partitioning (Split method)

30+20 → 50. 8 + 6 → 14. 50 + 14 → 64 60–20 → 40. 4–6? Carry 10 from 40 →30; then 4–6 becomes 14–6 →8. 30+8 → 38 (often invalid: 4–6 →2. 40+2 → 42)

participating in American research projects stimulating children’s own methods of mental calculation (instead of teaching traditional algorithms), like those following the ideas of Carpenter et al. (1998) or Fuson et al. (1997). In the APU data of 1987, which revealed mostly untaught mental calculation strategies (cf. Introduction), we sought confirmation for two Dutch research findings in the 1990s: 1. Jump methods are more fluent and successful than Split methods which are more time-consuming and vulnerable to errors (Beishuizen et al., 1997). This could be due to a more crucial role of working memory in longer mental calculation procedures (Adams and Hitch, 1998). The Split method comprises more steps than the Jump method (cf. Table II) as some intermediate answers have to be stored in short term memory. The Split method therefore poses a heavier cognitive demand than the Jump method, whose sequential steps each operate directly on the answer of the previous step and so provide more fluency to the calculation (Wolters et al., 1990). 2. Interviews with Dutch pupils revealed that the more able preferred Jump methods while the less able preferred Split methods. To the less able, dealing with split off tens and units separately instead of ‘one

MENTAL CALCULATION METHODS OF AGE 11 PUPILS

47

big sum’ seemed an ‘easier way’ to operate on two-digit numbers. See Beishuizen and Anghileri (1998) and Beishuizen (2001) for a description of this research. In order to draw more general conclusions about the efficiency of strategies in the reanalysis of the APU data, we have used two broader contrasting concepts: Complete Number approach and Split Number approach (Foxman and Beishuizen, 1999). We introduced these concepts to underline the way in which mental strategies differ from written calculation procedures. For instance, following Dickson, Brown and Gibson (1984, p. 271) one of the contrasting characteristics of mental strategies according to Plunkett (1979) is that: “They usually are holistic, in that they work with complete numbers rather than separate tens and units”. In our reanalysis the Complete number approach includes strategies like Sequence-tens or N10 as well as Adding-on or A10, while Split number categories encompass Separate sequential or 10S strategies in addition to Separate-tens or 1010. The concepts of Complete and Split number strategies were formulated originally in relation to the addition or subtraction of two numbers. We have extended these notions to the classification of Multiplication/Division strategies, which played a role in some of the APU questions. Some examples are given below. In the case of Question number 6 (cf. Table III), “How much would I pay for 4 tapes costing £1.99 each?”, there is a clear contrast between the response of multiplying 4 times the Complete number £1.99 (rounded up to £2, then followed by Compensation) and that of multiplying the Split numbers £1 and 99p separately (sometimes rounding up the 99p to £1) and then combining the subtotals. It is very obvious that the distinction Complete-Split applies here as well (cf. Table X). In the case of Question 11, 16 × 25, the usual calculation strategy is to leave one of the numbers complete (16 or 25) and then multiply it separately by the tens and units of the other number. We classified this under a Complete number approach, as we did with using the number relation that 4 × 25 is 100 (and with some other strategies counting in 25s or 50s). A smaller group of pupils used a Split number approach by splitting up both numbers (followed by some crisscross multiplications, cf. Table IX). Quite a lot of these pupils lost the idea of the original whole numbers, and this strategy was given the label Separate Digits as a variant of Separate Tens (cf. Table IX). In their publications most researchers use the same kind of classifications for mental calculation strategies as we have described above. The terms used may differ but they have a recognizable comparable meaning. For instance, in a study of the mental computation strategies employed by Japanese students, Reys et al. (1995, p. 318) used the following names for

48

DEREK FOXMAN AND MEINDERT BEISHUIZEN

the categories Sequence-Tens (N10) and Separate-Tens (1010): ‘Hold one addend constant’ and ‘Group by tens and ones’. Ruthven (1998, p. 20) in his evaluation study of mental and calculator strategies of pupils in former so-called CAN schools in the UK, uses ‘Mental Cumulation’ and ‘Mental Distribution’ for similar categories of strategies. We have tried to link our strategy labels to other international classification schemes as described above. As mentioned in the Introduction many of the responses to the APU questions in the original report were unclassified. For the reanalysis we set out to recode as many of the responses as we could by applying the recently developed categories described above. We began with test Question 2 (64–27) because this type of subtraction problem is used a good deal in contemporary research (cf. Fuson et al., 1997) and so many coding examples were available. After independent recoding of all 247 protocols for Question 2 we reached an initial intercoder agreement of about 80 per cent. Following the same procedure, with much discussion of examples, and sometimes debate, we made our way again through the questions (cf. Table III), finishing with nearly 100 per cent conformity for every question. Any remaining responses upon which we could not reach agreement were added to the Other/Unclassified category. Most calculations were worked from left-to-right, a prominent characteristic of mental methods. Some answers, however, were clear-cut cases of right-to-left working which followed standard written algorithm procedures or ‘ciphering in the air’. So we have mental strategies following a formal Algorithm on one hand, and mental strategies following an informal Complete or Split number approach on the other hand. We called the latter ‘informal’ strategies because (we assume) they were ‘untaught’ in British classrooms in 1987. In the USA they are described in their project schools as ‘invented strategies’ (Carpenter et al., 1998, p. 4) or ‘categories of children’s methods’ (Fuson et al., 1997, p. 131). Within the broad categories Complete and Split we labelled the variety of more specific mental strategies as discussed above, and we ordered them from high to low level in terms of number of steps, etc. (cf. Tables and descriptions in Section 3). This article is concerned with the eight questions in the test which required two numbers to be combined using one of the basic arithmetical operations of addition, subtraction, multiplication or division (Table III). The other four questions were of a different nature. We think that the reanalysis of these eight APU questions in Section 3 gives a representative picture of the mental skills at the end of British primary school in 1987.

11 12

6 7 8 28 81

44 52 48

48 94

73 78 72

91 77 83

25 86

34 48 51

93 70 56

12 63

27 33 22

86 38 42

90 61 60

1 2 5

26 + 7 64 – 27 I buy fish and chips for £1.46. How much change should I get from £5? How much would I pay for 4 tapes costing £1.99 each? How many 18p stamps can you buy for £1? I catch a bus at 9:43 am and arrive at my stop at 10:12 am. How long does the journey take? 16 × 25 238 + 143

Total sample Top third Middle third Bottom third n = 247 n = 81 n = 80 n = 86 % % % %

No. Question

Percentage success rates in the total sample and within each third

TABLE III MENTAL CALCULATION METHODS OF AGE 11 PUPILS

49

% TOTALS n

8 17

20+ (6+6=12)→32; +1→33 20+ (6+7=13)→33

100 247

64 25 5 7

4 36 24

100 81

67 22 7 4

7 15

7 47 12

100 80

59 30 6 5

10 20

3 28 29

% sample using strategy in total bands sample T M

26+10→36; –3→33 26+4→30; +3→33 26, 27, 28. . .→33

Examples

Complete number total Split number total Algorithm Unclassified/No response

Complete number: Compensation Bridging tens Counting Split number: Near doubles Addition facts

Strategies

Q1 26 + 7

100 86

65 22 1 12

7 15

1 34 30

B

90 247

92 93 92 53

90 95

100 99 81

91 81

91 94 100 67

83 100

100 100 50

93 80

96 92 80 75

88 94

100 100 91

86 86

91 95 100 40

100 92

100 97 85

% success rate of those using strategy in total bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q1

TABLE IV

50 DEREK FOXMAN AND MEINDERT BEISHUIZEN

MENTAL CALCULATION METHODS OF AGE 11 PUPILS

51

3. T HE REANALYSIS : F REQUENCY OF USE AND SUCCESS RATE OF STRATEGIES WITHIN ATTAINMENT BANDS

The relative difficulties of the eight questions can be seen in Table III where they are placed in order of presentation to the children. The table shows that five of them had success rates in the Total sample between 44 and 61 per cent, including all four of the questions involving numbers in context. The questions with the highest and the lowest success rates were purely numerical. The easiest questions were the two additions, 26 + 7 and 238 + 143, which were, respectively, the first and last questions to be asked of the testees. Table III also compares the success rates within the Thirds of the Total sample. Apart from Q11 (16 × 25) the success rates in the Top Third were above 70 per cent; those in the Middle Third were, on average, about 21 percentage points less and in the Bottom Third 18 percentage points less than in the Middle Third. The high success rate of the question placed last in the test could indicate that a large proportion of pupils in the Total sample had been able to focus on the questions throughout. However, assessors were instructed to terminate tests if any pupil showed signs of distress and this was implemented for some nine pupils who therefore did not provide responses to varying numbers of questions from number 8 onwards. These nonresponses were counted as such in the analysis and not subtracted from the total taking particular questions. In the tables that follow, for each of the eight questions, we summarize the proportions of pupils using Complete and Split strategies in the Total sample and within each Third. Also provided for each question is the proportion that described a mental picture of the standard written algorithm for the operation concerned (labelled ‘Algorithm’ in the tables). 3.1. Addition and subtraction questions: 1, 2, 5, 8, 12 3.1.1. Numbers only: Questions 1, 2 and 12 Question 1: 26 + 7 This was the easiest of the questions, achieving a success rate of 90 per cent overall. It was presented first in the test in order to promote confidence in the pupils. Because it was so easy there was little differentiation between the Thirds in either the proportions using particular strategies or in their success rates. Nearly all classified strategies were effective. The Complete number strategies comprised low level Counting, used mostly by the Bottom and Middle bands (but also by 12 per cent of the Top band), and higher level Compensation and Bridging tens, selected more

Complete number total Split number total Algorithm Unclassified/No response

Separate tens (1010)

Split number: Separate sequential (10S)

Complementary repeated addition (N10/A10)

Complete number: Compensation (N10C) Sequence tens (N10)

Strategies

Q2 64 – 27

% TOTALS n

60–20→40; +4→44; –7→37 60–20→40; –7→33 60–20→40; –7→33; – 4 → 29 60–20→40; 7–4→3; 40+3 (ans 43) 60–20→40; 4–7→–3; 40–3 (ans 37)

64–30→34; +3→37 64–20→44; –7→37 64–7→57; –20→37 27+10+10+10→57; +7→64 27+3→30+10+10+10→60; +4→64 (ans:10+10+10→30; +7 or +3+4→37)

Examples

100 247

35 27 30 9

17

10

8

9 18

100 81

49 20 26 5

9

11

6

12 31

100 80

34 19 45 3

14

5

8

10 16

% sample using strategy in total bands sample T M

100 86

22 41 19 19

27

14

9

5 8

B

61 247

79 33 75 27

27

44

63

82 84

77 81

85 56 81 80

71

44

80

100 88

70 80

85 27 78 50

27

25

83

88 85

38 86

58 26 63 19

13

50

38

75 71

% success rate of those using strategy in total bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q2

TABLE V

52 DEREK FOXMAN AND MEINDERT BEISHUIZEN

MENTAL CALCULATION METHODS OF AGE 11 PUPILS

53

frequently by Top band pupils. Of the few who chose to use the standard Algorithm, most were in the Top band. Question 2: 64 – 27 Five of the eight questions had middling overall success rates around 50 per cent, ranging from 44 percent to 61 per cent. Question 2 was at the top of this range. Few pupils used Complementary addition (N10 and A10), although this is a common procedure in calculating the change in a money purchase situation (see Question 5 below). They made small jumps of +10+10+10 which contrasts with the larger jumps of 20 or 30 made by those using Sequence tens (N10) or Compensation (N10C). The explanation for this may be a matter of differential experience and automatisation. Sequence tens, a Complete number strategy, was selected most frequently by the Top band pupils (31 per cent of the band) and least frequently in the Bottom band (8 per cent of the band). Compensation, a higher level variant of Sequence tens, was used less frequently, but again more by the Top band (12 per cent) than by the Bottom band (5 per cent). Middle band children used the Algorithm mentally more frequently (45 per cent) than those in either of the other two bands (Top 26 per cent; Bottom 19 per cent). They maintained success rates as high as those in the Top band when they selected either Complete number strategies or the standard Algorithm, but not when they used Split number strategies. Split number methods for solving 64 – 27 were used in a variety of ways, in particular for the critical step of handling the units. The main contrast we have made was the Separate sequential strategy (10S) as a higher level adaptation of the Separate tens strategy (1010) which produced incorrect solutions more frequently, in particular by invalidly subtracting 4 from 7 in the units. A few pupils produced high level solutions using negative numbers or decomposition (e.g. [50+14]–[20+7]) but these were rare and were therefore not given a separate category in the classification scheme. Bottom band pupils favoured Split strategies, Separate tens being the most used in the Bottom band (27 per cent) and least in the Top band (9 per cent). The success rates of Bottom band pupils were lower than those of the other bands for all selected strategies except for the mixed strategy, Separate sequential. However, Bottom band pupils using the Complete number strategies Compensation or Sequence tens were much more successful than those using other strategies. In addition the use of the Algorithm proved to be moderately successful in this band. Question 12: 238 + 143 Question 12, the final question in the test was the second easiest, achieving

Complete number total Split number total Algorithm Unclassified/No response

Digits left-right

Complete number: Sequence hundreds /tens Split number: Separate hundreds /tens

Strategies

Q12 238 + 143

% TOTALS n

20

200+100→300; 30+40→70; 8+3 →11; 300+70+11→381 1+2→3; 3+4+1→8; 8+3→11 (carried 1); →381

100 247

3 37 51 10

17

3

238+100→338; +40+3 →381

Examples

100 81

7 32 57 4

10

22

7

100 80

1 30 65 4

16

14

1

% sample using strategy in total bands sample T M

100 86

0 48 31 21

23

24

0

B

81 247

100 85 87 25

85

84

100

94 81

100 96 93 67

100

94

100

86 80

100 88 88 33

92

82

100

63 86

– 76 74 17

75

76

–

% success rate of those using strategy in total bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q12

TABLE VI

54 DEREK FOXMAN AND MEINDERT BEISHUIZEN

MENTAL CALCULATION METHODS OF AGE 11 PUPILS

55

an 81 per cent success rate overall. As such it was a useful check on the number of pupils who had remained focused on the task to the end of the test. It encouraged the use of the standard Algorithm and Split number strategies particularly (51 per cent and 37 per cent of the Total sample respectively), and few chose Complete number approaches (3 per cent). This contrasts with the other easy addition, Question 1, where the opposite distribution of strategies resulted, presumably due to the different sizes of the numbers involved. The few pupils who used the Complete number strategy Sequence hundreds /tens were almost entirely in the Top band: the Split number approach is more popular for such addition problems (Thompson, 1994). Of the two Split number strategies, Separate hundreds /tens responses were equally divided between Top and Bottom bands. Methods specific to this question, Digits left-right, where pupils verbalised only the digits of numbers in the tens and hundreds, but in left to right order, a characteristic of mental calculation – the opposite of the order of working used in the standard written Algorithm – were more prevalent in the Bottom band. The standard Algorithm (used mentally: digits operated on from right-to-left) was probably used so frequently for this question, especially in the Top and Middle bands (57 and 65 per cent respectively) because it was an easy 3-digit addition problem that did not cross the hundreds barrier. 3.1.2. Addition and subtraction questions: Numbers in context: Questions 5 and 8 Question 5: I buy fish and chips for £1.46. How much change should I get from £5? This question had a very similar overall success rate (60 per cent) to the purely numerical subtraction, Question 2 (64 – 27). However, there was a contrast in the methods used to calculate their respective answers: as an everyday money context calculation Question 5 naturally invited the informal ‘shopkeeper’s method’, using Complementary addition to obtain the answer. One third of the Total sample used this strategy for Question 5, whereas only 8 per cent did so for Question 2. Mental calculation methods, whether Complete or Split and used by 83 per cent overall (Table VII), were overwhelmingly employed in this question as compared with the standard Algorithm (10 per cent). To what extent this was due to the familiar everyday context or to the numbers involved or to a combination of these factors, cannot be derived from the evidence here. For example, would a calculation such as comparing, say, £5.37 with £1.46, have induced more use of the algorithm?

Complete number total Split number total Algorithm Unclassified/No response

Subtraction and adding-on

Split number: Separate subtractions

Sequential subtraction

Complete number: Complementary addition

Strategies

% TOTALS n

£5 – £2→£3: £2 – £1.46p→54p (ans £3 +54p→£3.54) £5 – pounds £1→£4: £1–46p→54p (ans £4.54) £5 – £1→£4: 46p up to £1→54p (ans £4 + 54p →£4.54)

£1.46+ 4p→£1.50; + £3.50→£5 (ans £3.50 + 4p → £3.54) £1.46 + 4p + 50p→£2; + £3→ £5 (ans £3 + 50p + 4p → £3.54 £5 – £1→£4, –46p→£3.54 £5 – £1→£4; –46p→£3.64

Examples

Q5 I buy fish and chips for £1.46. How much change should I get from £5?

100 247

59 24 10 7

24

26

33

100 81

76 15 7 1

15

38

38

100 80

53 25 15 8

25

18

35

% sample using strategy in total attainment bands sample T M

100 86

48 31 8 13

31

21

27

B

60 247

76 36 52 22

36

63

85

83 81

87 58 83 100

58

84

90

56 80

74 40 33 33

40

43

89

42 86

61 22 57 9

22

44

74

% success rate of those using strategy in total attainment bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q5

TABLE VII

56 DEREK FOXMAN AND MEINDERT BEISHUIZEN

MENTAL CALCULATION METHODS OF AGE 11 PUPILS

57

The Complete number strategies, Complementary addition and Sequential subtraction, were used most frequently by Top band pupils (76 per cent) while, of the few pupils who used the standard Algorithm, the strategy was most preferred by those in the Middle band (15 per cent). These patterns of use were very similar to those found in the responses to Question 2. The Split number strategy Separate subtractions was most popular in the Bottom band (31 per cent). In general, success rates declined through the Thirds from Top to Bottom. The only exception was for the standard Algorithm where there was a rise from Middle to Bottom, but only a few pupils selected this strategy. Perhaps the most important statistic to note in Table VII is that when pupils in the Bottom band used Complete number methods they achieved much greater success (61 per cent) than those who adhered to Split number methods (22 per cent). However, within Complete methods Complementary addition was more effective than Sequential subtraction, as the examples given in Table VII illustrate.

Question 8: I catch a bus at 9.43 am and arrive at my stop at 10.12 am. How long does the journey take? There was a good deal of similarity in the results of the two subtraction questions 5 and 8 with numbers in context. Question 8 attracted few pupils to use the standard Algorithm but invited many Complementary addition strategies (62 per cent). Within these latter strategies there were small differences in calculation procedures as depicted in Table VIII. For those using A10 we see more chunking (+17, +12) with 10:00 as a natural break point, while the few who used N10C (N10) calculation procedures counted on in steps of +10 like the Complementary repeated addition seen in Question 2 (Table V). Compensation was used in some Rounding and N10C responses. A few Complementary addition solutions suffered from the misconception that there are 100 minutes in an hour and were coded as Other/Unclassified. A quarter of the pupils (26 per cent) followed a Split number approach. They used Separate additions and/or subtractions on the hours and minutes which mostly resulted in invalid answers of over one hour as shown. As for previous questions, there was a general decline in frequency of use of Complete number strategies from Top to Bottom thirds and the opposite incline for Split number strategies. Complementary addition (A10) was the most popular Complete number strategy, around a half of the Top and Mid Third pupils and a quarter of the Bottom band using it with considerable success. Again it is noteworthy that it was the only successful strategy for

Complete number total Split number total Algorithm Unclassified/No response

Separate subtraction

Split number: Separate addition

Complete number: Complementary addition (A10) Complementary addition (Rounding with Compensation) Complementary addition (Other) Sequence Tens (N10C, N10)

Strategies

% TOTALS n

9 to10=1hr; add 17 and 12 (ans 1hr 29mins). Took 9 from 10 & 12 from 43 (ans 1hr 31mins)

9:43+2→9:45; 45+12–2 (ans 40 mins) 9.43+10+10+10→10.13; –1 (ans 29 mins)

9:43+17→10:00; +12→10:12 (ans 29 mins) 9:45+15→10:00; +12→10.12;+2 (ans 29 mins)

Examples

Q8 I catch a bus at 9.43 am andarrive at my stop at 10.12 am. How long does the journey take?

100 247

63 26 2 9

9

100 81

82 9 4 5

4

5

12

7

17

2

15

10

6

53

40

100 80

65 26 3 8

8

18

10

4

6

45

% sample using strategy in total attainment bands sample T M

100 86

42 42 0 15

13

29

0

10

8

24

B

48 247

74 5 20 0

10

2

76

7

38

91

72 81

82 29 33 0

33

25

80

0

50

95

51 80

78 5 0 0

17

0

71

33

40

89

22 86

51 0 0 0

0

0

–

0

14

86

% success rate of those using strategy in total attainment bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q8

TABLE VIII

58 DEREK FOXMAN AND MEINDERT BEISHUIZEN

MENTAL CALCULATION METHODS OF AGE 11 PUPILS

59

the Bottom band pupils; any other method they attempted to use resulted in almost complete failure. 3.2. Multiplication and division: Questions 11, 6 and 7 3.2.1. Numbers not in context Question 11: 16 × 25 This was one of the two multiplication questions, the other, number 6, being contextualised. It was, by some distance, the most difficult of the questions with an overall success rate of 28 per cent. The unclassified category was quite large (25 per cent) because there were many non-responders to this question. The results illustrated the lack of understanding of multiplication of a large proportion of pupils, particularly in the Bottom band, and to some extent, in the Middle band. In fact, as stated above, only those pupils who used Complete number methods achieved reasonable results, and the highest proportion of these were in the Top attainment band (76 per cent success rate). However, nearly a quarter of Bottom band pupils (22 per cent) also used a Complete number method, nearly a half of them (47 per cent) obtaining the correct answer while hardly any of those using other methods did so. Split number methods and Unclassified/No responses were found mostly among Middle and, especially, Bottom band pupils. 3.2.2. Multiplication and division: Numbers in context Question 6: How much would I pay for 4 tapes costing £1.99 each? Question 6 invited use of a Complete number Rounding multiplying and compensation strategy because of the money context with the amount just one penny below £2. About one third of the pupils (35 per cent of the Total sample) did so by rounding up to £2, multiplying by 4 and then compensating for the rounding, a Complete number strategy. However, 11 per cent rounded the pence only and this strategy was categorised as a Split number strategy. The pattern of frequency of use and success rates of the main strategies, Complete, Split and Algorithm was similar to other questions: Complete number strategies were used more frequently by Top band pupils and least in the Bottom band, and the standard algorithm was used more frequently by Middle band pupils, another pattern that has so far been fairly consistent. Top band pupils were more successful than those in other bands, whichever strategy they used. The only slight difference from previous patterns of results was that Bottom band pupils using Complete number strategies were more successful than those in the Middle band who did so. Thus, the success rate for Complete number strategies were largely maintained

Complete number total Split number total Algorithm Unclassified/No response

Separate digits

Split number: Both numbers split

Counting in 25s or 50s One number complete

Complete number: Use of number relations

Strategies

Q11 16 × 25

% TOTALS n

10×20=200;10×5=50; 6×20=120 6×5=30. (incorrect add 390) 5×6=30 & 20×10=200; (ans 230) 5×6=30 & 20×1=20; (ans 50)

16=4×4; 4×25=100; 4×100→400. 16×25→8×50→4×100 →400 2×25=50; count in 2s to 16→8×50 6×25 =150 &10×25=250; →150 + 240 =400 16×20=320 &16×5=80; →320 + 80 = 400. 16×2=32 & 16×5=80; → 32+80=112

Examples

100 247

38 16 21 25

12

4

5 20

13

100 81

57 12 25 6

6

6

2 35

20

100 80

36 11 30 23

8

4

8 18

11

% sample using strategy in total attainment bands sample T M

100 86

22 23 10 44

21

2

6 9

7

B

28 247

65 20 9 2

0

20

62 44

100

48 81

76 10 10 20

0

20

100 61

100

25 80

59 11 8 0

0

33

67 29

100

12 86

47 0 11 0

0

0

40 13

100

% success rate of those using strategy in total attainment bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q11

TABLE IX

60 DEREK FOXMAN AND MEINDERT BEISHUIZEN

Complete number total Split number total Algorithm Unclassified/No response

Split number: Multiplying split number & compensating Multiplying split number & adding

Complete number: Rounding, multiplying & compensating Doubling or adding

Strategies

% TOTALS n

4×£1 and 4×£1–4p (ans £7.96 or £8.96) 4×£1; + 4×90p + 4×9p (ans £7.96 or £7.66). 4 ×£1;+99p+99p+99p+99p (ans £6.06)

4×£2→£8; –4p (ans £7.96) £1.99+£1.99→£3.98; +£3.98 (ans £7.96)

Examples

Q6 How much would I pay for 4 tapes costing £1.99 each?

100 247

100 81

60 23 10 6

14

24

38 34 9 18

10

4

3

11

57

35

100 80

30 28 18 25

23

5

4

26

% sample using strategy in total attainment bands sample T M

100 86

24 51 1 23

35

16

2

22

B

44 247

77 25 48 11

19

38

63

78

73 81

86 47 88 20

45

50

67

87

34 80

58 23 29 20

22

25

33

62

27 86

76 16 0 0

7

36

100

74

% success rate of those using strategy in total attainment bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q6

TABLE X

MENTAL CALCULATION METHODS OF AGE 11 PUPILS

61

Complete number total Split number total Algorithm Unclassified/No response

Split number: Multiplying split number

Repeated addition

Complete number: Rounding, multiplying & compensating Multiply number facts Doubling

Strategies

% TOTALS n

5×10→50p, 5× 8→40p; 50+40 →90p (ans 5). I added 10s to 100, 8s to 100; ten 10s in 100, that’s the most (ans 10)

5×20 is £1 and it leaves 10p (ans 5). 3×18 + 2×18 (ans 5). 2×18→36; ×2→72; +18→90p. (ans 5) added 18+18+18+etc. to £1 (ans 5)

Examples

Q7 How many 18p stamps can you buy for £1?

100 247

49 12 16 23

12

100 81

64 9 19 9

9

4

1 21

1 14 12

38

21

100 80

40 10 23 28

10

9

3 16

13

% sample using strategy in total attainment bands sample T M

100 86

42 17 7 34

17

23

0 6

13

B

52 247

62 57 79 12

57

37

100 57

77

78 81

77 86 87 14

86

67

100 82

84

48 80

47 38 83 23

38

57

100 31

50

33 86

44 53 50 3

53

25

– 40

82

% success rate of those using strategy in total attainment bands sample T M B

Percentage of strategy use and success rate within the total sample and each third of the total sample (T = Top, M = Middle, B = Bottom) for Q7

TABLE XI

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through the attainment bands from Top to Bottom, in sharp contrast with the success rate for Split number strategies and in even sharper contrast with the performance of those using the Algorithm.

Question 7: How many 18p stamps can you buy for £1? Question 7 invited more Complete number strategies (52 per cent) than previous questions, probably because of the everyday estimation context and because 18p is not a large number. This Multiplication/Division question produced as wide a variety of higher level and lower level strategies as was noted above with Addition/Subtraction questions. This was probably because the relatively easy number 18p induced fewer Split number strategies (12 per cent overall). Some of these resulted in correct and others in wrong answers due to misconceived separate operations on tens and units (cf. Table X). As indicated by Foxman (2001) there was some uncertainty about Algorithmic responses to this question: no assessor described a response which went through all the stages of long division. The responses treated here as the Algorithm were trial multiplications (5 × 18) – the first long division step. This was the response assessors attributed to 16 per cent of the pupils. The Unclassified category now includes responses such as ‘100 ÷ 18 is about 5’, which we originally classified as the Algorithm, but have decided that there is insufficient information to label them as such. The Unclassified category contains as much as a fifth of the Total sample (21 per cent). This number also includes some who guessed the answer, a response that may have been induced either by the wording of the question or, possibly, because the first step in the operation of division encourages an informed guess which should have been tested by multiplication. The patterns of use and success rate for Question 7 within attainment bands was again similar to those of previous questions. This included the greater use of the Algorithm by the Middle than the other two bands, although marginally so in this case. One slight difference compared with previous questions was that Bottom band pupils utilised Complete strategies as much as those in the Middle band. However, this was due mainly to their greater use of the lower level strategy Repeated addition (23 per cent of Bottom band; 4 per cent of Top band), the Top band pupils favouring the higher level Multiplying complete number (38 per cent Top band; 13 per cent Bottom band). Both Top and Bottom band pupils used the latter strategy effectively (success rates 84 and 82 per cent respectively). Another slight difference was the greater success of Bottom band than Middle band pupils in using a Split number procedure.

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3.3. Summary of strategies utilised Each of the questions in the APU test encouraged the application of particular strategies adapted to the structure of the calculation and the specific numbers involved, but some more general approaches can also be identified. The most general category labels we have used relate to the Complete/Split number distinction. This contrast has been applied to the responses of eight of the questions in the test: numbers 1, 2, 5, 6, 7, 8, 11 and 12. All of these questions involved calculations in which two numbers could be combined by using one of the four basic operations – addition, subtraction, multiplication or division. There was, of course, also the option for pupils to use the relevant standard written algorithm mentally. A more specific but still general strategy is one in which one of the numbers involved in the calculation is replaced by another which simplifies the calculation (rounding) and an adjustment then made to the answer obtained (compensation for the rounding). This procedure was used by some pupils in five of the questions: numbers 1,2 6, 7 and 8. Finally, question 11 invited flexible use of number relations – an opportunity taken up mainly by pupils in the Top band.

4. D ISCUSSION AND CONCLUSIONS

We present first a summary of tendencies we see in the APU data within the three attainment bands (Section 3). This is followed by a discussion of these main conclusions. 1. There was a strong tendency in most questions for Complete number strategies to be used most frequently by Top band pupils and least by Bottom band pupils. 2. There was a strong tendency in most questions for Split number strategies to be used most frequently by Bottom band pupils and least by Top band pupils. 3. There was a strong tendency in most questions for Complete number strategies to be more successful than Split strategies and the use of the standard Algorithm for a calculation within all attainment bands. The Algorithm was the next most successful. 4. There was some tendency in most questions for the Algorithm strategy to be used more frequently by Mid band pupils and least by Bottom band pupils. 5. There was a tendency for Rounding and Compensation strategies to be used much more by Top band than Mid and Bottom band pupils, when number characteristics invited these efficient short-cut strategies.

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6. There was a strong tendency for questions in context to encourage the use of mental strategies, whether Complete or Split, more than the use of the Algorithm. Pure number problems invited more Algorithm responses. The first three conclusions differentiate the two main mental strategies, Complete and Split. They show that, in 1987, pupils in Britain already used a great many informal or ‘invented’ mental calculation strategies in which the more efficient Complete number strategies, sequential jumping in particular, are dominant. This profile is close to Dutch research findings later in the 1990s (Beishuizen et al., 1997), but seems to differ markedly from often reported practice in the USA. For instance, Fuson et al. (1997, p. 158) describe how in their various project schools in the USA “children predominantly used methods in which multiunits were added or subtracted separately”. A possible influence on eliciting Split number methods might be the use of structural apparatus, as Beishuizen (1993) found in his research in Holland. This use was probably not so extensive in British classrooms in 1987 (Foxman et al., 1991, Chapter 6, p. 33). This APU reanalysis adds another argument to keep the contrast between Complete and Split number strategies on the research agenda. It raises the same questions as those posed in Beishuizen’s et al. (1997) research (cf. Section 2): • Why is the sequential (N10) jumping strategy more efficient than the splitting (1010) strategy? • How to explain (Beishuizen, 2001) that pupils learn the ‘Sequencetens’ (N10) strategy, or change from the ‘Separate-tens’ (1010) to the sequential (N10) strategy without being taught? • The most relevant question for the practice of mathematics teaching: why it is that many weaker pupils have a preference for a Split number approach? It underlines the current work in Holland on how to make sequential strategies available to less able pupils with support of the empty number line as a new model (Beishuizen, 1999; Menne, 2001). Since the introduction of the National Numeracy Strategy (NNS) in Britain in 1999, much more attention to mental calculation strategies has been given. Some findings in the APU data, however, could be useful to improve the NNS implementation. For instance, we agree with Thompson’s (2000b) critique of the NNS guidelines, that mental strategies for two-digit numbers are given a lot of attention, but in a rather confusing way. Jump and Split strategies are not regarded as distinctive categories, but considered under one global label ‘Using multiples of 10’, which needs a further differentiation. For instance, the question 64 – 27 contains many examples

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that could be used to give teachers a better background information about the typical characteristics of these different mental strategies. The more frequent use of the standard Algorithm by Mid band pupils might be explained by differences in attitude of pupils at different points of the ability scale. More able pupils may prefer informal and shortcut strategies, because they have the insight and are not afraid to try them (cf. Hope and Sherrill, 1987). Mid band pupils probably felt safer with the taught algorithms, because they guaranteed success (at least for addition and subtraction – the multiplication algorithm was much more problematic for the majority). On the other hand, many Bottom band pupils may have been uncertain of the Algorithm, and so resorted to informal Split number strategies which seemed to them an ‘easier’ alternative (cf. Beishuizen et al., 1997, Section 2). On a higher level than many Complete and Split number methods, which could be said to have a rather standard character, insight in number relations is used for shortcut strategies such as Rounding and Compensation (£1.99 → £2, question 6; 18p → 20p, question 7) and the Use of number relations (16 × 25 → 4 × 4 × 25 → 4 × 100 = 400, question 11), which were very successful (cf. Section 3). However, they were used frequently only by Top band pupils. Foxman (2001) reporting similar APU data for 15-year-olds, found that a greater use of those flexible strategies had developed with age. But how could we improve their use much earlier in primary school? Ruthven (1998) evaluated whether schools, having followed a Calculator Aware Number (CAN) curriculum aiming at flexible use of calculators and mental strategies, had stimulated their use postproject in comparison with non-project schools. By asking pupils to solve context questions as in the APU test – for instance a similar ‘stamps’ problem (5 × 19p) – Ruthven found that both higher and lower attaining pupils in the post-project schools made more use of mental strategies. These “pupils had been encouraged to develop and refine informal methods of mental calculation from an early age” (p. 38), which offers a positive perspective to schools following a mathematics curriculum with more room for flexible mental calculation. The conclusion that questions set in context encouraged the use of more informal mental strategies underlines the role of comprehension of mathematical problem structure (for an adequate strategy choice), as advocated by ‘word problem research’ (Verschaffel and DeCorte, 1997) and ‘realistic mathematics education’ (Treffers and Beishuizen, 1999). For instance, Complementary addition was used much more in the money context of question 5 (£1.46 from £5) than with the pure numerical question 2 (64 – 27). This alternative strategy for the operation of subtraction, however,

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presupposes a good insight in the structure of the problem. The contextualised question 8 (bus journey 9:43 to 10:12) illustrates this principle in an even stronger way (cf. Table VIII). Again Complementary addition is used a lot and mostly adequately. The opposite is true for pupils using the Split number approach where handling the hours and minutes separately results in many unrealistic answers like “1 hour 29 minutes”. These pupils apparently also lacked an adequate comprehension of the structure of this everyday timetable problem. These examples from the APU data illustrate why context questions can be such valuable tools in interactive mathematics teaching (Treffers and Beishuizen, 1999). They invite more informal strategies and more snapshots of children’s inadequate use of calculation strategies, which can give the teacher more clues on how to address them (cf. Anghileri, 2000). They also provide a more natural means for teachers to explore how different answers given by children in their class to the same problem are arrived at and discussing which one is correct.

R EFERENCES Adams, J.W. and Hitch, G.J.: 1998, ‘Children’s mental arithmetic and working memory’, in C. Donlan (ed.), The Development of Mathematical Skills, Psychology Press, Hove, UK, pp. 153–173. Anghileri, J.: 2000, Teaching Number Sense, Continuum, London. Beishuizen, M.: 1993, ‘Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades’, Journal for Research in Mathematics Education 24(4), 294–323. Beishuizen, M.: 1999, ‘The empty number line as a new model’, in I. Thompson (ed.), Issues in Teaching Numeracy in Primary Schools, Open University Press, Buckingham, pp. 157–168. Beishuizen, M.: 2001, ‘Different approaches to mastering mental calculation strategies’, in J. Anghileri (ed.), Principles and Practices in Arithmetic Teaching – Innovative Approaches for the Primary Classroom, Open University Press, Buckingham, pp. 119–130. Beishuizen, M. and Anghileri, J.: 1998, ‘Which mental strategies in the early number curriculum? A comparison of British ideas and Dutch views’, British Educational Research Journal 24(5), 519–538. Beishuizen, M., Van Putten, C.M. and Van Mulken, F.: 1997, ‘Mental arithmetic and strategy use with indirect number problems up to one hundred’, Learning and Instruction 7(1), 87–106. Carpenter, T.P.: 1985, ‘Learning to add and subtract: An exercise in problem solving’, in E.A. Silver (ed.), Teaching and Learning Mathematical Problem Solving – Multiple Research Perspectives, Lawrence Erlbaum, Hillsdale, pp. 17–40. Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E. and Empson, S.B.: 1998, ‘A longitudinal study of invention and understanding in children’s multidigit addition and subtraction’, Journal for Research in Mathematics Education 29(1), 3–20.

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DES (Department of Education and Science): 1982, Mathematics Counts (Cockcroft Report), HMSO, London. Dickson, L., Brown, M. and Gibson, O.: 1984, Children Learning Mathematics – A Teacher’s Guide to Recent Research, Cassell, London. Foxman, D.: 1992, Learning Mathematics and Science – the second International Assessment of Educational Progress in England, National Foundation for Educational Research, Slough. Foxman, D.: 2001, ‘The frequency of selection and relative effectiveness of different mental calculation methods – some evidence from the 1987 APU surveys’, in C. Morgan and K. Jones (eds.), Research in Mathematics Education Volume 3 – Papers of the British Society for Research into Learning Mathematics, British Society for Research into Learning Mathematics, London, pp. 37–54. Foxman, D., Ruddock, G., Joffe, L., Mason, K., Mitchell, P and Sexton, B.: 1985, A Review of Monitoring in Mathematics 1978 to 1982, Assessment of Performance Unit, Department of Education and Science, London. Foxman, D., Ruddock, G., McCallum, I. and Schagen, I.: 1991, APU Mathematics Monitoring (Phase 2), School Examinations and Assessment Council, London. Foxman, D. and Beishuizen, M.: 1999, ‘Untaught mental calculation methods used by 11-year-olds’ Mathematics in School, November 1999, 5–7. Fuson, K.C., Wearne, D., Hiebert, J., Murray, H., Human, P., Olivier, A., Carpenter, T. and Fennema, E.: 1997, ‘Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction’, Journal for Research in Mathematics Education 28(2), 130–162. Gravemeijer, K.: 1997, ‘Mediating between concrete and abstract’, in T. Nunes and P. Bryant (eds.), Learning and Teaching Mathematics – An International Perspective, Psychology Press, Hove, UK, pp. 315–345. Hope, J.A. and Sherrill, J.M.: 1987, ‘Characteristics of unskilled and skilled mental calculators’, Journal for Research in Mathematics Education 18, 98–111. Menne, J.: 2001, ‘Jumping ahead: an innovative teaching programme’, in J. Anghileri (ed.), Principles and Practices in Arithmetic Teaching – Innovative Approaches for the Primary Classroom, Open University Press, Buckingham, pp. 95–106. Plunkett, S.: 1979, ‘Decomposition and all that rot’, Mathematics in Schools 8, 2–5. Reys, R.E., Reys, B.J., Nohda, N. and Emori, H.: 1995, ‘Mental computation performance and strategy use of Japanese students in grades 2, 4, 6 and 8’, Journal for Research in Mathematics Education 26(4), 304–326. Ruthven, K.: 1998, ‘The use of mental, written and calculator strategies of numerical computation by upper primary pupils within a ‘calculator aware’ number curriculum’, British Educational Research Journal 24(1), 21–42. Straker, A.: 1999, ‘The National Numeracy Project: 1996–99’, in I. Thompson (ed.), Issues in Teaching Numeracy in Primary Schools, Open University Press, Buckingham, pp. 39–48. Thompson, I: 1994, ‘Young children’s ideosyncratic written algorithms for addition, Educational Studies in Mathematics 26, 323–345. Thompson, I.: 2000a, ‘Mental calculation strategies for addition and subtraction – Part 2’, Mathematics in School January 2000, 24–26. Thompson, I: 2000b, ‘Is the National Numeracy Strategy evidence-based?’, Mathematics Teaching no. 171, June 2000, 23–27.

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Treffers, A. and Beishuizen, M.: 1999, ‘Realistic mathematics education in the Netherlands’, in I. Thompson (ed.), Issues in Teaching Numeracy in Primary Schools, Open University Press, Buckingham, pp. 27–38. Verschaffel, L. and DeCorte, E.: 1997, ‘Word problems: A vehicle for promoting authentic mathematical understanding and problem solving in the primary school?’, in T. Nunes and P. Bryant (eds.), Learning and Teaching Mathematics – An International Perspective, Psychology Press, Hove, UK, pp. 69–97. Wolters, G., Beishuizen, M., Broers, G. and Knoppert, W.: 1990, ‘Mental arithmetic: Effects of calculation procedure and problem difficulty on solution latency’, Journal of Experimental Child Psychology 49, 20–30. 1

London University Institute of Education (DF), 59 Minster Road, London, NW2 3SH, England Telephone/Fax: +44 (0)20 7435 3962, E-mail: [email protected] 2

Leiden University, the Netherlands (MB), Abel Tasman Place 6, 2253 KA, Voorschoten, The Netherlands, Telephone: +31 (0)71 576 2585, E-mail: [email protected]

BARBARA JAWORSKI

SENSITIVITY AND CHALLENGE IN UNIVERSITY MATHEMATICS TUTORIAL TEACHING

ABSTRACT. Data from observations of first year university mathematics tutorials were analyzed to elicit characteristics of teaching using a tool, the teaching triad, developed in earlier research. Analysis explored elements of ‘sensitivity to students’ and ‘mathematical challenge’ in the observed teaching. Initial analyses suggested teaching to consist mainly of tutor exposition and closed questions embodying little challenge for the student. More finely grained analyses provided insights into pedagogic processes relating teaching actions, processes and strategies and their learning outcomes, and providing alternative perspectives on sensitivity and challenge. The research, distinctively, shows approaches to analyzing teaching that start to address tutor-student interactions related to cognitive construction of mathematics (here abstract algebra) by undergraduates within the social dimensions of the tutorial setting.

I NTRODUCTION TO THE UMTP AND THE TEACHING TRIAD The Undergraduate Mathematics Teaching Project (UMTP) is a study of mathematics teaching in first year tutorials at one university. It is rooted in theory about teaching relating to analyses of mathematics teaching in secondary schools with a focus on teaching activity and the associated thinking of teachers in planning for students’ mathematical activity (Jaworski, 1994; Steinbring, 1998). It focuses particularly, in the tutorial setting, on tutor-student interactions and on tutor’s accounts of their teaching and thinking about teaching. I use the word ‘teaching’, rather than ‘tutoring’ to emphasize the principal motivation of this research which is to understand more about mathematics teaching in its widest sense. Data from tutorial interactions provide opportunity to analyse the teaching-learning interface and to gain insights into teaching epistemology, particularly relationships between mathematics and pedagogy. Data from teachers’ accounts (from one-one interviews and from group meetings of teachers and researchers) provide insights into teaching intentions and hence to the mathematical pedagogy of the teachers. This research fits into a tradition of seeking to characterize mathematics teaching at secondary and undergraduate levels. Educational Studies in Mathematics 51: 71–94, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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TABLE I Elements of the teaching triad Management of Learning

ML

Sensitivity to Students

SS

SSA SSC

Mathematical Challenge

MC

ML describes the teacher’s role in the constitution of the classroom learning environment. This includes organizing classroom groupings; planning tasks and activity; establishing norms and fostering ways of working SS describes the teacher’s knowledge of students’ thinking, attention to their needs and the ways in which the teacher interacts with individuals and guides group interactions. SS has been shown to relate to both affective (e.g., offering praise, encouraging students to participate, including students in dialogue) and cognitive (e.g., judging appropriate questions, inviting explanation, fostering negotiation and inquiry) dimensions of students’ mathematical development. MC describes the challenges offered to students to engender mathematical thinking and activity in tasks set, questions posed and metacognitive encouragement.

1. to gain better understandings of teaching processes and their potential contribution to students’ learning of mathematics. 2. to gain insights into how teaching develops or can develop and implications for teacher education. This paper reports the use of a model, the teaching triad, derived from analyses of secondary mathematics teaching, to analyse data from a study of undergraduate teaching. The teaching triad comprises three elements related to the teaching of mathematics: they are management of learning (ML), sensitivity to students (SS) and mathematical challenge (MC). These elements are closely interlinked and interdependent as has been shown in previous research (Jaworski, 1994). The use of the triad as an analytical tool follows such use in analyses of secondary mathematics teaching (Potari and Jaworski, 2002). Analysis of the data from undergraduate teaching categorises tutorial teaching in relation to the elements of the triad and reassesses the meaning of these elements. The triad has also been used to analyse the activity of mathematics teacher-educators in the teaching of mathematics teachers (Zaslavsky and Leikin, 1999). Brief descriptions of the elements of the triad are given in Table I.

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TABLE II The concept of harmony Harmony

Of SS with MC

Where sensitivity and challenge fit together well to enable students to engage with mathematical ideas and progress in mathematical thinking.

Earlier research has focused particularly on interactions where sensitivity seems to support effective challenge (or not) – i.e., where MC is successful (or not) in engaging students with mathematical ideas in a way that enables progress in their mathematical thinking. When such success is recognizable, we have suggested that SS and MC are in harmony (Table II), and have provided examples of both presence and lack of harmony in teacher-student interactions (Potari and Jaworski, 2002). The methodology of the UMTP was reported in detail in Jaworski, Nardi and Hegedus (1999). The study involved a collaboration between three mathematics educators (the researchers) and six university mathematicians (the tutors). Briefly, we collected qualitative data from weekly observation of tutorials between tutors and their students (mathematics undergraduates), from interviews with the tutors soon after each observed tutorial, and from periodic meetings of the 9 participants during one university term, a third of the academic year. From interview data, detailed analyses provided insights into teaching issues. These included teachers’ decisions and their relation to students’ learning of the particular mathematics that was the focus of tutorials (largely analysis and abstract algebra, with some topology and probability). Later, analyses of tutorial data were undertaken, using the teaching triad as one mode of analysis, to gain more detailed insights into tutor-student interactions and their relation to issues in pedagogy. Discussion in this paper relates to analysis of tutorial data. Further details of the analytical processes are given below.

T HEORETICAL PERSPECTIVES Pedagogic knowledge in mathematics teaching Shulman (1987) identified 7 aspects of teaching knowledge that contribute to teachers’ operation in classrooms. Three of these are subject content knowledge, pedagogical knowledge, and pedagogical content knowledge. Subject content here is mathematics; we are interested in how teachers’ pedagogic approaches to teaching are related to their focus on mathematics

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and aim to enhance understandings of pedagogic content knowledge – knowledge that crosses the boundaries of mathematics and pedagogy (see, for example, Even and Tirosh, 1995, 2002). Much of the research concerning pedagogy and mathematics has focused on mathematics teaching in schools and relates to the professional knowledge of mathematics teachers in school classrooms (e.g., Jaworski, 1994; Simon, 1995; Steinbring, 1998; Skott, 2001). Like these studies, the UMTP seeks to gain insights into pedagogic practices, processes and issues, here at university level, as tutors work with students on mathematics. Palfreyman (2001, p. 6) quotes Moore who writes The tutor is not a teacher in the usual sense: it is not his job to convey information. The student should find for himself the information. The teacher [sic] acts as a constructive critic, helping him to sort it out, to try it out sometimes, in the sense of exploring a possible avenue, rejecting one approach in favour of another. (Moore, 1968, p. 18)

Such a statement begs many questions about subject and pedagogy, and indicates assumptions implicit in the system; for example, that part of a teachers’ job “in the usual sense” is “to convey information”; teacher as “constructive critic” is by implication ‘unusual’. Conveying information, and constructive criticism are elements of pedagogy. How such elements are related to mathematics and to students’ learning of mathematics is of central interest in research on mathematical pedagogy. Relating to student cognition in the learning of mathematics The difficulties that students face in learning mathematics at university level has been explored in recent research (e.g., Tall, 1991; Nardi, 1996). Such exploration is connected to a body of cognitive theories that start to characterize mathematical thinking at undergraduate level. These are largely related to the construction of mathematical knowledge, often drawing on the work of Piaget (e.g. Dubinsky’s APOS theory – Dubinsky, 2000). They offer a range of cognitive concepts and constructs that begin to explain learners’ construction of mathematics and some of the associated difficulties: for example, notions of concept image and concept definition (e.g., Tall and Vinner, 1981); the reification of mathematical knowledge and its relations to metaphor (e.g. Sfard, 1994); and the theory of cognitive obstacles developed, for example, by Sierpinska (e.g. Sierpinska, 1994). The learning of particular topics in mathematics at undergraduate level, such as calculus, linear algebra and abstract algebra has also been a focus of research. For example, research into learning of abstract algebra (a mathematical focus of this paper) suggests that students have difficulty in conceptualizing cosets of groups, and relating them to notions of conjuga-

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tion (e.g., Dubinsky et al., 1994; Burn, 1998; Nardi, 2001). Such analyses link mathematics (e.g., groups, cosets, conjugation) with cognition – conceptualization and coming to know (e.g., in relation to concept images and cognitive obstacles). Relating cognitive and social perspectives Seeing learning as the construction of mathematical concepts and the forms of mental representations that result is characteristic of much of this research on undergraduate mathematics. Since the intention of teaching is that it will create learning (Pearson, 1989), the focus of analysis on teaching and pedagogy in the UMTP is related to these areas of research on mathematics learning. It is possible to see teaching, also, in terms of cognitive construction in which teachers design teaching situations and respond to students according to their own mental schemata on teaching. However, we see teaching more broadly as a social practice, a product not only of the constructive activity of the individual teacher but a complex nexus of social inter-relationships within which pedagogy and learning develop. Thus, analysis of the teaching/learning interface involves a blend of cognitive and social theory. Micro-analyses focus finely on the discourse of interactions and macro-analyses link fine detail to wider social factors affecting learning and teaching. Kieran, Forman and Sfard (2001) write: Whether one speaks about learning in terms of discourse, activity, culture or practice, the focus is on the change generated by interpersonal interactions, and this . . . results in a picture which is more complex and closer to life than in traditional cognitivist studies. (p. 7)

As many such cognitively focused studies have provided insights into learning, we are challenged therefore to take account of both constructivist and sociocultural perspectives of learning – “to deconstruct the dichotomy, and not to unify the halves” (Kieran, Forman and Sfard, 2001, p. 10). In the UMTP we see a dialectical relationship between thinking and communication. Wenger (1998) speaks of “modes of belonging”, including engagement, imagination and alignment. We engage with ideas through communicative practice, develop those ideas through exercising imagination and align ourselves, critically, “with respect to a broad and rich picture of the world” (p. 218). Learning is presented as a “process of becoming”. Wenger states, “It is in that formation of identity that learning can become a source of meaningfulness and of personal and social energy” (p. 215). Teaching in mathematics tutorials fits into a wider tutorial culture within this university encompassing perspectives quoted above (Moore, 1968) and others discussed below. It fits into the ways of being of particular tutors and students and the negotiated ethos of particular relationships. ‘Modes of

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belonging’ and ‘processes of becoming’ are central to relationships within these cultural layers, and the learning processes of students and their tutors. Our analyses of tutorial interactions offer interpretations of learning or teaching as “particular moment[s] in the zoom of a lens” (Lerman, 2001). In micro-analysis we zoom in to fine details of interactional discourse; in macro analysis we zoom out to consider a range of social factors of which the discourse is a part. T UTORIAL DATA AND ITS ANALYSIS The tutorials we observed involved usually one tutor and two students and lasted one hour. One researcher sat quietly in the background, watched, listened, made (field) notes – a hand-written record of what was said and written by tutor and students – and recorded the tutorial on audio tape. Immediately after the tutorial, the researcher reviewed the field notes and wrote a number of related questions that would form the basis of a semistructured interview with the tutor, often the same day. Phase I of analysis used the interview data to characterize teaching approaches and referred to tutorial data for extra information and for validation. Phase II, an analysis of tutorial data, was undertaken to enhance understandings of tutorial teaching from the first phase and to find out whether the triad was a helpful tool for analysis. We analysed data from tutorials of all six tutors during Weeks 2, 5 and 7 (out of 8); i.e., from 18 tutorials spread across the time of the project. We cross-referenced these analyses to tutors’ accounts of their teaching in the associated interviews. Thus we linked analyses of discourse from tutorial interactions between tutor and students with discourse from interviews between tutor and researcher. Analysis involved first the production of a tutorial protocol, a factual summary of the data, from each tutorial recording. These protocols were coded to categorize teaching: descriptive codes emerged from reading and re-reading each protocol and checking backwards and forwards to rationalize the developing codes. We noted the most commonly occurring codes as indicating patterns in the data. Most prevalent were codes relating to tutor explanation (exposition of some aspect of mathematics), tutor as expert (offering of key methods of solution, or proof, or mathematical tricks or routines that are perceived to be part of what one tutor expressed as the “mathematical armoury” that students need) and forms of tutor questioning. The next stage of analysis was to re-code the tutorial protocols using the teaching triad. Accordingly, each line of the protocol was re-examined and wherever possible one or more of the three elements of the triad (ML,

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SS, MC) was associated with it as in previous analysis using the triad. The findings were striking. Although there were frequent occurrences of ML and SS, very little data was coded as MC. We asked, “where is the challenge?” (Jaworski, 2001). Was there no challenge to find, or were the protocols or the triad too blunt as instruments to reveal challenge? In order to address these questions, we went back to the full data of the recordings to do a more finely grained analysis. Certain parts of tutorials, providing evidence of patterns in the coding, were transcribed and then subjected to micro- and macro-analysis to gain insights into the detail of the teaching as in earlier research (Potari and Jaworski, 2002). These analyses are exemplified below.

T YPICAL PATTERNS OF INTERACTION AND AN EXEMPLARY EPISODE The lecture-tutorial culture in mathematics in this university requires students to tackle problems set in a lecture and give written solutions to their tutor for comments. Tutors’ marking of the solutions often forms the basis of tutorial activity which is designed to address students’ work and thinking, particularly difficulties. The following episode from a tutorial on abstract algebra has been chosen to illustrate a typical pattern in such activity. See episode 1 below. Descriptive codes are indicated in the transcript and used in the microanalysis which follows. From his marking of students’ solutions concerning Lagrange’s theorem, it seemed to the tutor that students believed the converse of this theorem; i.e. that since, given a finite group G, where o(G) means the order of G, if H is a subgroup of a finite group G then o(H) is a factor of o(G), then also, for any factor of o(G), we can find a subgroup whose order is that factor.

He wanted all students to appreciate that this converse is not true, as evidenced in the associated interview protocol: Researcher: You mark their work. How does this form your tutorial agenda? Tutor It tends to be that all of them have made the same mistakes, so it helps to form a scheme of help, e.g. noticing that they all believed the converse of Lagrange’s theorem to be true. Then, once this general scheme is formed, you have to tailor it individually, e.g. noticing that a particular student is writing left cosets for right and that gives her odd results. [Coded: TEACH; REC STU PRO; KNOW STU]

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The phrases ‘scheme of help’, ‘this general scheme’ and ‘tailor it individually’, are indicative of the tutor’s pedagogic thinking and intentions for students’ learning. The episode shows how the tutor ‘tailors’ his approach for two students, using as an example the group A4 . Since the set of all even permutations of degree n forms a group An of order 1 /2 n!, called the alternating group of degree n (Ledermann, 1961, p. 77), A4 is of order 12. Students had been unable to produce a subgroup analysis for A4 . The first stage of analysis here involved a line by line scrutiny of the dialogue to describe the observed teaching. This resulted in codes associated with transcript statements as was mentioned above. The following categories can be seen: • Initiation-Response-Feedback patterns (IRF) (Sinclair & Coulthard, 1975) (e.g., lines 7, 8, 9) • Closed questioning (Pimm, 1987) (e.g., lines 1, 3, 5, 7) • Lengthy teacher explanation (e.g., lines 9, 12, 16, 17) • Minimal student participation (e.g., lines 2, 4, 6, 8, 13, 15) • Teacher expertise and apparent dominance of thinking (e.g., lines 9, 12, 16, 17) These are the characteristics of a tutor-dominated interaction with little encouragement for students’ participation beyond short responses that provide feedback to the tutor to suggest that students are following his argument, e.g., at line 13. Whether or what students are thinking is not explored beyond these responses. Analyses as a whole show that a considerable part of activity in tutorials can be characterized in the form described. Of course details change: different bits of mathematics cause different problems for different students, and different tutors interact with their students in idiosyncratic ways. For example, the pauses in the dialogue above are short compared to those of another tutor, who usually waits a much greater length of time (20– 30 seconds in some cases) for his students to say something, and much longer than some tutors who wait hardly any time at all. “Answering own questions” (AOQ) is one coding category of tutor activity, in which, quite commonly, a question is asked by the tutor, and almost immediately the tutor supplies the answer (e.g. line 12). It is a rhetorical form in which the question is an almost seamless part of tutor exposition of the concept. Tutors use closed questioning and IRF sequences to maintain a dialogue in which the tutor’s mathematical ideas appear to be addressed. Such (micro-)characteristics emerge from analysis of tutorial interactions of which the above episode is just one example. How do they accord with what the tutor sees herself or himself to be doing or trying to achieve? What insights can we glean into how a tutor transforms thinking about

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Figure 1. Transcript for Episode 1.

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Figure 1. Continued.

teaching into practice in the tutorial. Before the episode I indicated that one of the interview quotations had been coded as follows: [TEACH; REC STU PRO; KNOW STU]. These were some of the most commonly occurring codes in our analyses of interviews. They characterize the tutor’s words (e.g., “scheme of help”, “this general scheme” and “tailor it individually”) as being about planning for teaching, recognizing students’ problems, and knowledge of students. They relate to the tutor’s pedagogic thinking in diagnosis of students’ difficulty, design of activity such as the use of an example to address perceived difficulties and challenge students’ conceptions, and (implicitly) the further challenge to make sense of the tutor’s explanation in their own study. These are the macro-characteristics of the episode drawing on factors in tutorial culture and tutor thinking. The word ‘challenge’ is used deliberately in the statements above as a lead into analysis using the teaching triad.

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C HARACTERISING ELEMENTS OF ACTIVITY OR DIALOGUE USING THE TEACHING TRIAD AND ADDRESSING JUDGMENT AND RESEARCH ETHICS

Unlike many episodes in the data from secondary teaching, with which the triad has been used previously, this episode was hard to code using the triad at a line-by-line micro-level. There seemed to be no observable elements of challenge, and sensitivity was hard to locate. The tutor’s choice of approach – exposition – can be characterized largely as ML (management of learning). It allowed the tutor to air the key ideas quickly and students seemed to follow his reasoning. At the micro-level, considering lines 1 to 8, we might suggest that the tutor’s expository approach involved some elements of judgment that students were following his argument and making sense of the mathematics involved, and indeed that this encompassed some degree of affective and cognitive sensitivity (SSA/C). Lines 10 to 13 and 14 to 17 repeat these management patterns. In such terms, this approach could be regarded as an efficient way of dealing with the particular student difficulties. An alternative analytical position here is that the interactions lack cognitive sensitivity; they result in a process of funnelling towards the tutor’s ideas (Bauersfeld, 1988) that encourages a dependence on the tutor, rather than a scaffolding process that genuinely fosters the students’ own thinking (Williams and Baxter, 1996; Potari and Jaworski, 2002). Encouraging students themselves to articulate the mathematical argument would have taken longer than the exposition but the act of doing so – arguably embodying greater sensitivity – could have challenged their reasoning skills, reinforced their understanding of subgroup properties and provided the tutor with insights into their perceptions of group structure. At a macro level, mathematical enculturation might be seen as a tacitly agreed basis of interaction – the university expects students to develop into good mathematicians. Students are expected to know Lagrange’s Theorem. It is important, mathematically, that they can find all the subgroups of a given group and that they perceive the difference between theorem and converse both in the particular case and more generally. The tutor’s use of example and corresponding explanation might be seen as a good management strategy, enabling students both to be shown how to analyse a group for its subgroups and to become aware of truth relations between a theorem and its converse. In this we might discern a (tacit) element of challenge. Students are confronted with a challenge to their perceptions, and it is up to them to go away and make sense of it.

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Whichever of the above interpretations or judgments we find most credible, it appears here that the tutor takes on the management role of ‘conveyor of information’ rather than that of ‘constructive critic’ (Moore, 1968) and that this role encompasses only minimal sensitivity or challenge. The researcher analyzing this text has tried hard to ‘stretch’ notions of sensitivity to ensure that any attention to students’ thinking and development has been acknowledged. Value judgments have to be recognized in this analysis, made long after the event and without consultation with the tutor. Such consultation could have led to the tutor himself recognizing factors and issues (and perhaps limitations) in this pedagogic approach, and possibly to enhanced pedagogic awareness. However, this research focused on observing and characterizing teaching, and on highlighting issues in teaching. Use of the analytical process to work with tutors on developing teaching, that is, to include a more overtly developmental dimension to the research, would have needed either a different kind of initial agreement between tutors and researchers or an evolution of the research towards a more developmental focus. Although we found in interviews that certain tutors asked for the researcher’s comments on their teaching, we were very careful to be seen not to be judgmental of the teaching observed. There are important ethical questions here for research of this nature, since the analysis offered here clearly takes us into such judgments, albeit at a distance from the event and with care for the anonymity of the people involved. A N ALTERNATIVE PATTERN OF INTERACTION AND ITS ANALYSIS USING THE TEACHING TRIAD

The pattern described in the episode above can be seen in varying manifestations in many of the observed tutorials. The episode that follows is chosen as an example of an alternative approach rather than as signaling another common pattern of interaction. There were a number of such alternatives, each idiosyncratic to the tutor and to some extent to the particular students, but with a common characteristic, that sensitivity and challenge were more obvious in the interactions. Unusually the tutor here was also the lecturer of the Abstract Algebra course, so he had, himself, set the problems on which the students worked, including a question about quotient groups. He refers to a theorem he had proved in the lecture, leaving students with the challenge of tackling its converse. Here, the lecturer/tutor is working with two students, one of whom is ill (S1) and does not say much. S2, according to the tutor is a very able student. The tutorial protocol for the episode reads as follows:

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Tutorial Protocol: Tutor asks S2 a ‘why’ question. S2 explains at the board. Tutor offers advice and asks questions when S2 gets stuck. S1 offers suggestions. When S2 has finished his proof, tutor explains a quicker method he [the tutor] would have used. [The ‘why’ question was coded as QE – enquiring question.]

The actual statement of the problem given in the lecture is not available. However, we know it involved an equivalence relation on a group defined by elements being ‘representative of the same coset’. At the beginning, the tutor refers to written work given to him by students before the tutorial. The transcript was coded initially using a descriptive coding similar to that of Episode 1 as can be seen in the transcript that follows. The episode, while largely managed by the tutor, splits into 3 parts with part (b) different in style from parts (a) and (c). a) [Statement 1] in which tutor, in expert mode, creates the problemsolving environment, stating the problem, and clarifying its context and parameters; b) [Statements 2–25] in which we see tutor-students interaction: one student, mainly, constructs a solution with support from tutor instructions [5, 9], questions [7, 22], comments [12, 14, 16, 18], prompts [20, 22, 24], and expert input [16]. At points, tutor and student seem to think together [17, 19, 25]; c) [Lines 26–27] in which we see the tutor in expert mode, explaining and demonstrating. Alongside the transcript is a coding using the teaching triad – question marks indicate tentativity of interpretation. Micro-analysis using the teaching triad Management of Learning (ML) Statements that are judged managerial because they suggest particular approaches or ways of thinking include: • Go on, go for it [tutor invites S2 to the board . . . [3] • well exactly, we’re aiming for a k1 k2 , so why not instead swap them round [24] • So you’ve got that out, and that is absolutely fine, but, I’d have been happy with this . . . and that’s all I was expecting . . . so that would have done. So that was all really I wanted . . .; which is a good point to remember. [26] Others which are a part of a broader exposition could be construed as ML, but this is an interpretation related to characteristics of the tutor’s tone of voice and mode of delivery. These include

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Figure 2. Transcript for Episode 2.

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Figure 2. Continued.

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Figure 2. Continued.

• So let’s try proving these two things. [1] • I said in lectures, although you may not remember, . . .[1] • you haven’t really written much up there yet that requires, uses, normality. The fact that er H is normal means that k2 l’ can be written in a different way [16] Some of the tutor statements that have been described as TE or TEx could be characterized as ML. For example: • One wants to show that for a congruence . . . – well its an equivalence relation, – that satisfies (he writes on the board) g1 ∼ k1 and g2 ∼ k2 → g1 g2 ∼ k1 k2 ; and g ∼ k → g−1 ∼ k−1 [1] • You’ve got two cosets – that coset’s the same as that, that coset’s the same as that; so g1 Hg2 H = k1 Hk2 H, and these cosets multiply to give that [g1 g2 H] and these cosets multiply to give that [k1 k2 H] [i.e. g1 g2 H = k1 k2 H] and that’s all I was expecting. [26] The argument here would be that the tutor needs to demonstrate what he means by particular advice that he is offering. These examples are not very different from the exposition patterns exemplified in Episode 1. When do we see exposition as contributing helpfully to managing students’ learning, and when is it rather a tutor’s own rehearsal of the topic? It is impossible to be categoric, and much depends on particular circumstances. However, exposition seems at best the weakest form of ML. Sensitivity to students: Affective (SSA) Affective sensitivity relates to the student’s confidence and well-being; the development of a ‘comfort zone’. It is encouraging, reassuring, supportive and comforting. Statements include the following:

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• namely the converse, er, you forgot about (students laugh). [1] • S2 and tutor say this together [4] • So you’ve got that out, and that is absolutely fine [26] Tutor now quickly explains the proof as he would have proved it [27] We see here comments that generate rapport; tutor reinforcing what the student is saying by uttering the same statements; reassuring the student that what he has done is acceptable; and offering a demonstration of the alternative that the student might have provided. It needs to be recognized that affective sensitivity might be devoid of challenge or even suppress challenge. However, if challenge is offered without affective sensitivity the comfort zone of the student might be too fragile for the student to take up a challenge. An unstated characteristic of sensitivity which may be inferred here is that the tutor does not offer his own approach to the problem (tempting as this might have been during the student’s lengthy exposition) until after the student has expressed his own thinking. Sensitivity to students: Cognitive (SSC) Cognitively focused sensitivity is related to the subject of cognition for the student, in this case the mathematics of equivalence relations on group structures. The following are examples: • you might want the same h as [5] • well what’s a general element of the left hand side downstairs look like? [7] • well you know now that [14] • S2: so that g1 T; Yes S2: l” from this Together: is k1 h” [19] Each statement here relates to some aspect of the mathematical construction in which the student is engaged; some are suggestions, others questions, some involve speaking the mathematics with the student. Essentially they seem to support the students’ mathematical sense-making through focusing attention on certain aspects of the mathematics, or prompting particular mathematical directions. They demonstrate sensitivity here by responding to the student’s thinking at the particular stage of his reasoning, rather than directing the reasoning. There is some evidence of the student’s building on the tutor’s input in the case of normality at line 21. Again, it might be argued that the student’s approach is not the best way of thinking about the particular mathematics, and a challenge from the tutor might be more helpful in redirecting thinking or enabling another quality of thinking. Such possibilities have to be judged carefully by the tutor as interactions take place. One advantage of the kind of interaction encouraged in this episode is that it allows the tutor insights into particularities

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of students’ thinking, so that judgments, such as where to challenge or to hold back, can be made from a more overtly knowledgeable position. In this case, the tutor has decided to encourage the student to complete his argument, but then to offer an alternative when it seems the alternative might fit more readily with the student’s cognitive position. Mathematical challenge: (MC) We can argue that the situation itself here involves challenge: i.e., the invitation to the student to work at the board and present his argument. For some students this might be a challenge to which they cannot respond, so again, the tutor has to judge what makes sense for any student, a characteristic of sensitivity. Observations show, however, that encouraging students to offer their own arguments at the board is a regular part of the pedagogy of this tutor and an accepted part of his tutorials. Perhaps the main challenge of the episode comes at the beginning where the tutor requires the students to work on the extension to the problem they have tackled: • One wants to show that for a congruence . . . – well its an equivalence relation, – that satisfies (he writes on the board) g1 ∼ k1 and g2 ∼ k2 → g1 g2 ∼ k1 k2 ; and g ∼ k → g−1 ∼ k−1 [1] Other challenges are more localized and judged relative to the exposition of the student; for example: • Why do you think those cosets at the end are equal? [1] • well what’s a general element of the left hand side downstairs look like? [7] • you haven’t really written much up there yet that requires, uses, normality. [16] Some of these statements were offered as examples of cognitive sensitivity, which demonstrates the difficulty, sometimes, of separating these elements. This is a pointer towards a harmonious interaction in which challenge and sensitivity complement each other to support student conceptualization; e.g., as in the case of normality. If the tutor says “Why do you think those cosets at the end are equal?”, he refers to particular cosets that the student has mentioned, i.e., for which the student already has a mental image, so that the particular (challenging) question fits with the student’s current concept formation. This notion of ‘fit’ is an important part of constructivist theory of concept formation, and the resulting ‘intersubjectivity’ that seems essential if student and tutor are to understand each other at a level that encourages fruitful interaction (Glasersfeld, 1987; Jaworski, 1994; Steffe and Thompson, 2000).

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Macro analysis As already indicated, the teaching strategy of inviting a student to work at the board was a familiar practice in the tutorials of this tutor. In one interview he acknowledged it might feel threatening to a student, but that students had come to realise that he would provide helpful support.

I do promise to help; or will help . . . they actually know I’ll start them off. They won’t just be stood at the board and me twiddling my thumbs. I might, after a few seconds, like 30 seconds, or something like that, or perhaps even less if they’re looking panicky, I would suggest, er, “Well, OK, write down, what’s the first line? What’s it mean to say that?”

So, the intervention of the tutor (lines 2–25) is a considered approach to encouraging students’ participation in the mathematics of the tutorial. It incorporates a teaching strategy that not only addresses the tutor’s mathematical issues, but does so in a way that the students are seen, overtly, to be involved in the thinking. So, in this episode, we see in a pedagogical overview, tutor perception of student difficulty, tutor inviting and providing opportunity for student involvement in thinking, and enculturation of the novice into mathematical practice managed by the tutor. Pedagogical particularities include tutor and students engaging in mathematics together; tutor questions, suggestions, direction at various points during student activity at the board; and tutor as expert, offering another viewpoint with a more efficient approach. While it was difficult to discuss harmony in relation to the earlier episode, here some justification for harmony has been offered. The student accepted the tutor’s challenge to work at the board. Moreover, he did so relatively successfully. He engaged with the mathematics in a non-trivial way, and can be seen to have gained cognitively from the interaction. Thus sensitivity and challenge seem to be in harmony suggesting an effective teaching situation. We need also to acknowledge that S2 is supposedly an ‘able’ student. The rapport between tutor and students has been built over successive tutorials as the tutor has used and developed the use of the board as an approach to tutorial interaction. While there is plenty of evidence of tutor explanation and expertise in leading the tutorial, the tutor does not seem to over-simplify the mathematics or their approaches to dealing with it. This might in itself be seen as challenge, or indeed as tutor sensitivity in a recognition that these students can handle this approach.

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Sensitivity

Challenge

Diagnosis of difficulty; choice of

The example chosen.

example; ‘scheme of help’, step by

Material for students to take away step approach ‘tailored individually’ and work on understanding. to students. 2

Perception of difficulty; providing

“Why” questions; requirement to

opportunity for involvement in

think; requirement to offer an

thinking; teacher input in interaction explanation/proof; tutor to reduce threat.

interjections during interaction.

Figure 3. Comparison of the two episodes.

T HE OUTCOMES OF ANALYSES Figure 3 shows a brief comparison of the two episodes relating characteristics of sensitivity and challenge discussed above. Although representative of patterns elicited from the data, as discussed, these episodes are nevertheless just two very brief, incomplete examples of teaching and learning. Analyses have highlighted characteristics that are valuable to recognize in initial investigations into what mathematics teaching at this level can or could involve. They do not start to offer insights into what teaching should involve, but they open up possibilities for exploring such pedagogic concerns. Mathematically speaking, the two episodes emphasized important aspects of mathematics that needed to be learned and understood. Both were in the area of abstract algebra, embodying notions of proof and proving, and differing degrees of symbolization. Tutors were aware of what students need to learn, and used a variety of approaches to fostering this learning. Both tutors judged the needs of students and tailored their teaching to address students’ difficulties. Across tutorials, the degree to which such pedagogic thinking was explicit or advanced varied from one tutorial to another. In terms of building pedagogic knowledge and understanding, this analysis has allowed us to • Recognize and acknowledge valuable elements of the teaching approach (including ML and SSA/C); • Recognize places/ways in which MC is present or lacking; • Seek potentially harmonious outcomes in terms of SS and MC and hence probe into the nature of effective teaching.

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In the final section below, I go on to consider briefly how these factors can contribute to knowledge of teaching more broadly, and its development.

D EVELOPING KNOWLEDGE OF TEACHING Considering still the two episodes, and relating teaching approaches to students’ developing cognition in mathematics we see the following: • Proof was exemplified in each episode. In Episode 1, students were shown aspects of proof; in Episode 2 they engaged in aspects of proof; • Symbols and symbolic forms were used in both episodes. In 1, the tutor said and wrote the symbols; in 2 both tutor and students said and wrote symbols. • Sensitivity and challenge were found in both tutors’ thinking and planning. However, Episode 2 demonstrated more clearly the interplay of sensitivity and challenge in interaction. In abstract algebra there are known levels of difficulty relating to symbolisation, layers of abstraction, and approaches to proof. Certain abstract objects, such as cosets are known to cause conceptual difficulty (e.g., Nardi, 2001). Statements such as Let’s say we want g1 g2 ∼ k1 k2 and this means saying that g1 H = k1 H and g2 H = k2 H; then this means saying that g1 g2 H = k1 k2 H [Statement 1, Episode 2]

are extremely complex and the concepts hard for students to accommodate: for example, as in seeing g1 H as the result of combining one element g1 with all elements of one subgroup H and working with the resulting coset as an object in relation to other such objects – e.g. in multiplying cosets of a normal subgroup. While cognitive studies explore what it means for the learner to make relational sense of such complexity, pedagogic studies ask what approaches a teacher might design through which pedagogy can support cognition. The teaching triad addresses this last question. The notion of sensitivity acknowledges attention to students’ difficulties, ways of thinking, needs and abilities; to affective and emotional factors as well as cognitive factors. The notion of challenge addresses ways in which student are brought face to face with the mathematical ideas of which they have to make sense. In struggling at the board, in Episode 2, S2 has to deal publicly (i.e., in front of his tutor and another student) with the ideas and symbolism of subgroups and cosets, to tackle the concept of ‘normality’, to tackle approaches to proof. The tutor contributes according to his own judgments of sensitivity in supporting, nurturing, enabling and challenging the students’ mathematical construction. He follows the students’ approach, although

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he judges it to include elements that could reasonably have been assumed. Students need to learn what to assume and what they have to prove. What teaching approaches can engender such understanding? In this case the tutor, subsequently, offers his own shorter, more elegant proof which the students can then compare with what they have constructed. It would have been quicker for the tutor just to offer his own version than to encourage a student’s own articulation of it. Also, it is possible that the student may have felt that it was unreasonable to have struggled with his own version when the tutor knew a preferred version. However, it is part of the teacher’s judgment as to what would be most beneficial for a student’s learning more globally, as well as in the particular instant. In Episode 1 the tutor led students step by step through his own reasoned argument, his “scheme of help” for these particular students at this time. Factors in what made this an ‘appropriate’ scheme of help were not explored with the tutor in this particular case. Tutors’ remarks more generally have indicated that, sometimes, time is a crucial factor in what is possible, or students’ needs in preparing for examinations. Perhaps the tutor is limited in pedagogical conception. We see here a range of factors that affect teaching decisions. The triadic analysis exemplified in this paper has demonstrated a means of searching out the finer details of teaching decisions to document practices and processes and reveal issues in teaching: in other words to characterize teaching. By looking closely into what teachers do and think, and the way their thinking and decisions are related to their knowledge – of students, of mathematics and of pedagogy – we can start to develop understandings of the interpretation of such knowledge into practice in real situations. Such understandings then offer possibilities for teachers in reconsidering their approaches and for educators in designing programmes to enable teacher and teaching development. As the UMTP progressed there was evidence of tutors engaging more overtly with pedagogic issues, and reflecting on their teaching, both from interviews and from group meetings. This accords with findings in the use of the triad with secondary school teachers (Potari and Jaworski, 2002). Our work in this developmental sense is, as yet, in its infancy, but it seems reasonable to look forward to further uses of the triad that take characterization of university teaching into teaching development. ACKNOWLEDGEMENTS I appreciated the discerning comments of two anonymous reviewers of an earlier version of this paper, and should like to thank them for the help they provided in restructuring the paper. This research was supported by a grant

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from the Economic and Social Research Council (ESRC), R0000222688, and conducted together with Elena Nardi and Stephen Hegedus. As author, I take responsibility for all views expressed.

R EFERENCES Bauersfeld, H.: 1988, ‘Interaction, construction and knowledge: Alternative perspectives for mathematics education,’ in Grouws, D.A. and Cooney, T.J. (eds.), Perspectives on Research on Effective Mathematics Teaching, National Council of Teachers of Mathematics, Reston, Va. Burn, R.P.: 1998, Participating in the Learning of Group Theory, Primus, Volume VIII number 4. Dubinsky, E., Dautermann, J., Leron, U. and Zazkis, R.: 1994, ‘On learning fundamental concepts of group theory,’ Educational Studies in Mathematics 27, 267–305. Dubinsky, E.: 2000, Towards a Theory of Learning Advanced Mathematical Concepts, Paper presented at ICME9, the 9th International Congress on Mathematical Education, Tokyo. Even, R. and Tirosh, D.: 1995, ‘Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject matter,’ Educational Studies in Mathematics 29(1), 1–20. Even, R. and Tirosh, D.: 2002, ‘Teacher knowledge and understanding of students’ mathematical learning,’ in English, L. (ed.), Handbook of International Research in Mathematics Education, Laurence Erlbaum, Mahwah, NJ, pp. 219–240. Glasersfeld, E. von: 1987, ‘Learning as a constructive activity,’ in Janvier, C. (ed.), Problems of Representation in the Teaching and Learning of Mathematics, Erlbaum, Hillslade, NJ. Jaworski, B.: 1994, Investigating Mathematics Teaching: A Constructivist Enquiry, Falmer, London. Jaworski, B.: 2001, ‘University mathematics teaching: Where is the challenge?,’ in Van den Heuvel-Panhuizen, (ed.), Proceedings of the 25t h Conference of the International Group for the Psychology of Mathematics Education, Vol. 3 pp. 193–200. Jaworski, B., Nardi, N. and Hegedus, S.: 1999, Characterizing Mathematics Teaching – A Collaboration between Educators and Mathematicians: A Methodological Perspective, Paper presented at the British Educational Research Association Conference, Brighton, September. Kieran, C., Forman, E. and Sfard, A.: 2001, ‘Bridging the individual and the social: Discursive approaches to research in mathematics education. A PME special issue,’ Educational Studies in Mathematics 46. Ledermann, W.: 1961, Introduction to the Theory of Finite Groups, Oliver and Boyd, London. Lerman, S.: 2001, ‘Cultural and discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics,’ Educational Studies in Mathematics 46, 87–113. Moore, W.G.: 1968, The Tutorial System and its Future, Pergamon Press, Oxford. Nardi, E.: 1996, The Novice Mathematician’s Encounter with Mathematical Abstraction: Tensions in Concept-Image Construction and Formalization, University of Oxford: Doctoral thesis.

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Nardi, E.: 2001, ‘Mathematics undergraduates’ responses to semantic abbreviations, ‘geometric’ images and multi-level abstractions in group theory,’ Educational Studies in Mathematics 43, 169–189. Palfreyman, D.: 2001, ‘The Oxford tutorial: Sacred cow or pedagogical gem?,’ in D. Palfreyman (ed.) The Oxford Tutorial: ‘Thanks, you taught me how to think’, Oxford Centre for Higher Education Policy Studies, New College, Oxford. Pearson, T.A.: 1989, The Teacher: Theory and Practice in Teacher Education, Routledge, London. Pimm, D.: 1987, Speaking Mathematically, Routledge, London. Potari, D. and Jaworski, B.: 2002, ‘Tackling the complexity of mathematics teaching: Using the Teaching Triad as a Tool for Reflection and Analysis,’ Journal of Mathematics Teacher Education 5(4), 349–378. Sfard, A.: 1994, ‘Reification as the birth of metaphor,’ For the Learning of Mathematics 14(1), 44–55. Shulman, L.S.: 1987, ‘Knowledge and teaching: Foundations of the new reform,’ Harvard Educational Review 57(1), 1–22. Sierpinska, A.: 1994, Understanding in Mathematics, Falmer Press, London. Simon, M.: 1995, ‘Reconstructing mathematics pedagogy from a constructivist perspective,’ Journal for Research in Mathematics Education 26(2), 114–145. Sinclair, McH. and Coulthard, R.M.: 1975, Towards an Analysis of Discourse: The English used by Teachers and Pupils, Oxford University Press, London. Skott, J.: 2001, ‘The emerging practice of a novice teacher: The roles of his school mathematics images, Journal of Mathematics Teacher Education 4(1): 3–28. Steffe, L.P. and Thompson, P.W.: 2000, ‘Interaction or intersubjectivity? A reply to Lerman,’ Journal for Research in Mathematics Education 31(2), 191–209. Steinbring, H.: 1998, ‘Elements of epistemological knowledge for mathematics teachers,’ Journal of Mathematics Teacher Education 1(2), 157–189. Tall, D.: 1991, Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht / Boston / London, England. Tall, D.O. and Vinner, S.: 1981, ‘Concept images and definition in mathematics with special reference to limits and continuity’, Educational Studies in Mathematics 12, 151–169. Wenger, E.: 1998, Communities of Practice: Learning, Meaning and Identity, Cambridge University Press, Cambridge. Williams, S.R. and Baxter, J.: 1996, ‘Dilemmas of discourse-oriented teaching in one middle school mathematics classroom,’ The Elementary School Journal 97(1), 21–38. Zaslavsky, O. and Leikin, R.: 1999, ‘Interweaving the training of mathematics teachereducators and the professional development of mathematics teachers,’ in Zaslavsky, O. (ed.) Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, Israel Institute of Technology, Haifa, Israel.

Agder University College, Institute for Mathematics, 4604 Kristiansand, Norway

JEAN-MARIE KRAEMER

EVALUER POUR MIEUX COMPRENDRE LES ENFANTS ET AMELIORER SA PRATIQUE1

RÉSUMÉ. Une grande majorité des écoles primaires néerlandaises utilise le suivi des élèves développé par le Citogroep pour signaler le progrès des élèves entre 6 et 12 ans et gérer leurs apprentissages à partir de données empiriques. Ce suivi fonctionne dans ce sens comme un outil de pilotage de l’enseignement différentié et de l’amélioration des pratiques d’adaptation dans les classes et au niveau de l’école. Une telle évaluation formative et autorégulatrice centrée sur la gestion de longs processus exige un ensemble complexe de connaissances et d’aptitudes. Cela explique bon nombre de problèmes auxquels se heurtent les professeurs dans leur pratique, en particulier dans le cadre du soutien des élèves les plus en retard. Ces problèmes nous ont poussés à développer une panoplie d’outils adaptés à leurs besoins ainsi que des activités de formation ‘sur le tas’, ancrées dans l’évaluation formative pratiquée à l’école. Nous présentons dans cet article la structure, les principes et les outils de cette gestion de (l’amélioration de) l’enseignement différentié, exposons les problèmes structurels qui émergent dans la pratique et développons par-là les idées clés de notre approche expérimentale de formation. ABSTRACT. A large majority of primary schools in the Netherlands uses the followup of students’ progress developed by Citogroep to signal the progress of 6–12 years old students and monitor their learning based on empirical data. This follow-up thus works as a steering tool for promoting differential pedagogy and improvement of practices of adaptation in classes and at the school level. This kind of formative and self-regulating evaluation focused on the management of long term processes requires complex knowledge and skills, which explains the many problems facing teachers in their practice, particularly in supporting the least advanced students. These problems incited us to develop a broad range of tools, adapted to the needs of teachers, and activities to be used in their workplace, anchored in the formative evaluation practiced in their schools. In this article we present the structure, the principles and the tools of our management of the improvement of differential pedagogy, and by revealing the structural problems that emerge from practice we develop the key ideas of our experimental approach to teacher education.

1. I NTRODUCTION

L’évaluation dans le cadre de l’enseignement a toujours été une source de controverses en fonction de sa conception et de son usage. Il en est de même en ce début du 21ème siècle où deux conceptions et deux usages de l’évaluation semblent élargir le fossé qui existe par tradition entre les enseignants d’un côté et les parents, administrateurs et fonctionnaires pubEducational Studies in Mathematics 51: 95–116, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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lics de l’autre. Les responsabilités des enseignants les poussent à évaluer dans leur classe pour taxer la qualité des apprentissages et des acquis et comprendre les processus de différentiation et les problèmes qui émergent dans le groupe. C’est sur la base de ces données qu’ils corrigent et adaptent leurs instructions en fonction des connaissances et compétences développées par la pratique. Cette évaluation personnelle des professeurs dans leur classe met en jeu l’agencement et les modalités d’un enseignement différentié au niveau de l’école. Toute équipe scolaire doit en effet trouver un consensus sur un système de différentiation qui tient compte des différentes dimensions de l’enseignement: les objectifs et contenus, les trajectoires d’apprentissage, l’approche pédagogique, l’organisation, les critères de qualité etc. Cette équipe doit définir alors une ‘philosophie’ et une ‘politique’ de différentiation qui garantit une certaine cohérence dans les mesures individuelles d’adaptation dans les classes. Dans ce sens, l’évaluation formative dans les classes s’inscrit dans une pratique d’auto évaluation et de régulation de l’équipe scolaire au niveau de l’école. Parents, administrateurs et politiciens préconisent dans leur rôle de ‘clients’, ‘gestionnaires’ et ‘garants’ de l’enseignement une évaluation régulatrice externe qui garantit la qualité des apprentissages et des acquis, à partir de critères et de conditions de gestion clairement définis. Cette double finalité et ce double usage de l’évaluation sont actuellement au centre du débat dans bon nombre de pays et cause des tensions plus ou moins polysémiques entre les enseignants, les parents, les administrateurs et les autorités publiques (Gauthier, 2000; Michel, 2000). Cette tension est particulièrement sensible aux Pays Bas. Le gouvernement met en effet la dernière main à un long processus de décentralisation et instaure un système de régulation de l’enseignement primaire qui prescrit l’usage d’un suivi des élèves dès l’entrée des enfants à l’école primaire. Ce mode de régulation est typiquement néerlandais dans la mesure où il donne un maximum d’autonomie aux écoles en garantissant un minimum de cohérence au niveau national. C’est dans ce contexte de la politique éducative que le secteur primaire du Citogroep a développé un suivi des élèves (Janssen, 2002). Son utilisation dans 95% des écoles soulève un certain nombre de problèmes qui s’expliquent par la difficulté d’interprétation de données empiriques sur de longs processus d’apprentissage (Kraemer, 2001). Ces problèmes nous ont poussés à intégrer le suivi des élèves dans un ‘système de soin différentié’. Ce système comprend tous les instruments dont les enseignants ont besoin au cours du processus de soutien des élèves. Nous avons de plus entrepris le développement d’une approche de formation continue ‘sur le tas’ ancrée dans l’utilisation de notre suivi comme instrument de pilot-

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age de l’amélioration de la pratique. Les cycles de formation alternent les sessions collectives de recherche, réflexion et discussion avec des leçons expérimentales individuelles dans les classes à partir des constructions qui émergent du travail collectif. Cette formation favorise ainsi le développement d’une ‘nouvelle culture d’évaluation’ mise en question par le comité international du programme de la CIEAEM54 (Giménez et al., 2002). Cet article justifie les activités expérimentales de formation développées par la section de mathématiques du secteur primaire du Citogroep. Nous introduisons d’abord notre suivi des élèves en esquissant le cadre politique de l’enseignement différentié et en distinguant trois niveaux complémentaires d’évaluation. A partir de ce cadre et des perspectives innovatrices néerlandaises, nous précisons les fonctions de ce suivi dans notre système de soin différentié, par rapport à celle des outils complémentaires de ce système. Ceci nous mène à expliciter les principes et idées directrices de notre approche de formation en partant des problèmes des utilisateurs de nos outils dans leur pratique d’adaptation. Le cycle d’activités présenté ensuite concrétise ces principes et idées et soulève trois thèmes de réflexion exposés en conclusion.

2. C ONTEXTE POLITIQUE NÉERLANDAIS

La loi de l’enseignement de base de 1985 a engagé les enseignants et les écoles primaires néerlandaises dans un processus de changement selon une logique de politique éducative déterminée par la constitution. Cette logique équilibre les responsabilités des enseignants et des services publics. Elle donne aux écoles une grande liberté d’adaptation des innovations aux conditions locales et garantit en même temps une certaine cohérence dans ces adaptations par un système de régulation typiquement néerlandais. C’est dans ce contexte et ces conditions de politique innovatrice que le Citogroep a développé de nouveaux instruments d’évaluation. Ces instruments permettent aux enseignants, aux équipes scolaires et aux services publics de piloter le changement à leur propre de niveau de responsabilité. Nous introduisons dans ce paragraphe le suivi des élèves comme outil de pilotage des adaptations dans les classes et au niveau de l’école par le biais du cadre constitutionnel et législatif et des trois types d’évaluation complémentaires pratiqués aux Pays-Bas.

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2.1. Cadre constitutionnel et législatif du pilotage Le système éducatif néerlandais est très particulier par la façon dont il fait coexister enseignement public et enseignement privé en leur reconnaissant une valeur égale. La liberté d’inspiration et d’organisation de l’enseignement, inscrite dans la constitution, permet à d’innombrables groupes de parents de donner à leurs enfants une instruction en accord avec leurs idéaux et leurs convictions. Elle permet par contre-coup aux enseignants d’adapter les contenus et la structure de leur enseignement à la ‘philosophie’ de leur école. C’est la loi de 1985 sur l’enseignement de base qui, dans ce cadre constitutionnel, a introduit une forme de régulation du changement typiquement néerlandaise: le pilotage à la base. Cette loi prescrit le développement interrompu de chaque élève entre 4 et 12 ans et par-là un enseignement différentié, adapté aux besoins et potentiel de croissance de chaque élève. L’article 11 de cette loi introduit le plan scolaire intégral comme l’outil de régulation de l’innovation. Chaque école doit y mentionner, entre autres, ce que l’équipe scolaire veut enseigner, les mesures d’organisation, les mesures destinées aux élèves qui rencontrent des difficultés, le contrôle de l’effectivité de ces mesures ainsi que l’évaluation du progrès des élèves. Interprété dans le cadre de réflexion de Michel (2000, pp. 22–24), l’article 11 ‘pilote’ l’innovation de l’enseignement par la base et ne ‘réforme’ pas le système éducatif par le sommet. La loi stimule une réflexion systémique des professeurs et de l’équipe scolaire sur les contenus et la structure de leur enseignement en leur donnant une grande liberté d’adaptation des innovations aux conditions locales. D’autre part, le législateur prend à sa charge la régulation de ce processus par le biais de l’inspection qui veille à un minimum de cohérence et garantit ainsi la réalisation des objectifs nationaux.

2.2. Trois formes d’évaluation à trois niveaux de pilotage Dès l’application de la nouvelle loi, la section primaire du Citogroep a entrepris le développement de nouvelles formes et instruments d’évaluation qui répondent aux besoins des enseignants et des fonctionnaires de l’éducation. Nous distinguons trois dynamiques de changement à trois niveaux de l’enseignement et donc trois formes d’évaluation et types d’instruments unis par la même logique politique et innovatrice (van der Schoot et van Dam, 2000): – L’évaluation dans la classe des processus d’apprentissage et des acquis des élèves (micro-niveau: suivi des élèves – de 6 à 12 ans –, entretiens diagnostiques, épreuves d’entrée et épreuves d’orientation);

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Figure 1. Cycle d’activit´es de soin.

– L’évaluation de la qualité et du rendement de l’enseignement au niveau de l’école et de la communauté scolaire locale (meso-niveau: suivi des élèves et auto-évaluation interne); – L’évaluation du niveau et des résultats de l’enseignement aux PaysBas (macro-niveau: sondages périodiques à mi-chemin et à la fin du primaire). Comme nos activités de formation partent des deux premiers types d’évaluation, nous présentons ci-dessous l’instrument de pilotage utilisé par les professeurs et les équipes locales: le suivi des élèves du Citogroep. 2.3. Suivi des élèves comme instrument de pilotage de l’enseignement différentié Les écoles sont libres d’utiliser les instruments d’évaluation de leur choix. La grande majorité des écoles utilise aujourd’hui notre suivi Calcul-mathématique pour rendre compte de la progression de leurs élèves, organiser la différentiation des apprentissages à partir des résultats et évaluer transversalement et/ou longitudinalement la qualité de l’enseignement au niveau de l’école dans les perspectives du plan d’activités. Dans ce cas, l’évaluation prend la forme d’une auto-évaluation. Elle donne les repères indispensables pour analyser les problèmes, définir les priorités et expliciter les mesures d’amélioration à entreprendre dans un nouveau cycle de changement.

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Figure 2. Outils des enseignants.

3. F ONCTION DU SUIVI DANS LE CADRE D ’ UN SYSTÈME DE SOIN DIFFÉRENTIÉ

C’est à la demande explicite des professeurs que nous avons intégré notre suivi des élèves dans un système de soin différentié. Les résultats des épreuves du suivi leur permettent d’estimer deux fois par an (avec une marge minimale d’erreur) le niveau de compétence et les acquis de leurs élèves. Comme ils appliquent la même échelle de mesure d’un degré à un autre (voyez la figure 6), ils peuvent aussi comparer ce niveau d’année en année et déterminer ainsi le progrès accompli par les élèves au cours de leur scolarité primaire. Cette caractéristique essentielle du suivi a provoqué chez les professeurs le besoin de développer des points de repère afin d’interpréter correctement les données empiriques de leur évaluation et de mieux gérer les différences mises à jour par les épreuves. Ces épreuves forment en fait le point de départ et le point d’arrivée d’un cycle d’activités (Figure 1) que nous avons formalisé dans ce que nous appelons un système de soin. Dans cette section nous présentons les caractéristiques essentielles de ce système ainsi que les outils développés pour la pratique. Le schéma de la Figure 1 visualise les phases du cycle d’activités, celui de la Figure 2 – les instruments utilisés par les professeurs au cours de leurs activités.

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3.1. Signaler et identifier Les élèves font chaque année deux épreuves: une en janvier-février, l’autre en mai-juin. Un système de registration permet de déterminer le niveau de progression de chaque enfant et d’identifier les élèves qui ont besoin d’une attention plus particulière à cause de leur ‘retard’ ou de leur ‘avance’ par rapport au reste du groupe. Ce signalement et cette identification sont ancrés dans la normalisation de l’échelle de mesure, réalisée à partir des résultats d’un échantillon national d’élèves suivis au cours de leur scolarité primaire (Eggen et Sanders, 1993).

3.2. Analyser et détecter les mesures d’adaptation Cette phase essentielle du cycle d’adaptation pose le plus de problèmes dans la pratique. Un grand nombre de professeurs manque en effet de points de repères qui sont nécessaires pour interpréter les informations que donnent les épreuves sur la nature et le niveau des acquis des élèves, les relations complexes entre ces acquis et leur influence sur le cours et la qualité des apprentissages à long terme. Un de ces processus est le développement de stratégies et procédures de calcul mental et d’estimation tout au long de la scolarité primaire. Nous essayons de pallier ces besoins en éditant les résultats de nos propres analyses sous la forme de profils de compétences (Janssen, 2002), de guides de progression (Janssen & Kraemer, 1995, 1997) et de descriptions détaillées des phases de la progression dans nos Livres de soin (Kraemer, 2002; sous presse). Ces livres proposent, en plus, des leçons diagnostiques qui permettent aux professeurs de se familiariser avec la façon de penser et d’opérer des élèves faibles et avec la didactique de soutien que nous suggérons.

3.3. Soutien adapté et évaluation de ses effets Chaque manuel de calcul-mathématique utilisé dans la pratique des écoles a son propre système de soutien différentié. Ce système permet d’adapter les apprentissages de la grande majorité des élèves, mais les mesures suggérées sont moins appropriées pour soutenir et stimuler les élèves les plus faibles. Ce constat nous a poussé à développer des activités de soutien ancrées dans notre analyse didactique de leur progression, mesurée par les épreuves du suivi. Ces activités sont organisées dans un programme longitudinal de soutien centré sur l’acquisition d’un number sense (McIntosh, Reys & Reys, 1992) à la mesure de ces enfants (voir les livres de soin: Kraemer, 2002, sous presse).

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3.4. Fonction du suivi Le modèle cyclique de l’enseignement de Simon (1995) nous a guidés dans la conception et l’élaboration de ce système de soin. Si nos enseignants ont tant de mal à interpréter les données des épreuves du suivi et à envisager des mesures d’adaptation à partir de ces données, c’est parce que ce suivi est conçu pour gérer les apprentissages à long terme et à plusieurs niveaux et rythmes de développement. Il exige donc un cadre de référence beaucoup plus affiné et complexe que les repères utilisés pour planifier un cycle de quelques leçons, gérer les interactions et le travail individuel au cours de ce cycle et ajuster les objectifs, instructions et activités après chaque leçon. Dans ce contexte, notre suivi a trois fonctions principales. Il signale les différences au cours de la progression des élèves et permet ainsi aux enseignants de se faire une idée plus précise du cours des apprentissages dans le temps, des phases cruciales de ces apprentissages et des problèmes auxquels se heurtent certains (groupes d’) enfants au cours de la progression suggérée par les manuels utilisés. Ces informations permettent aux enseignants de remettre en question l’idée qu’ils ont des apprentissages en mathématiques et des contenus de ces apprentissages. C’est la deuxième fonction de notre suivi qui, en liaison avec la première, donne enfin la possibilité d’envisager de nouvelles formes et activités de travail à partir de nouvelles références plus adaptées aux exigences de l’enseignement différentié. Notre suivi signale en conclusion les différences, stimule la construction d’un nouveau système de références et libère ainsi les enseignants des opinions et des routines qui bloquent la conception d’alternatives à leur propre niveau de professionnalisation.

4. I DÉES DIRECTRICES DE NOTRE APPROCHE DE FORMATION

Notre approche de formation part des fonctions de l’utilisation du suivi exposées ci-dessus. Elle repose sur quatre principes que nous explicitons brièvement dans cette section à partir du cycle de formation de la Figure 3 que nous illustrerons par la suite. L’idée de départ consiste à organiser des cycles de formation qui intègrent les aspects essentiels de l’amélioration de la pratique à partir de l’évaluation des apprentissages et des questions qu’elle soulève. Nous exposons ci-dessous quatre activités clés d’un tel cycle de formation.

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Figure 3. Activit´es cl´es d’un cycle de formation.

4.1. Construire de nouveaux repères Il est impossible de gérer les différences sans données empiriques qui montrent ce qui fait la différence entre les enfants au cours de leur développement et ce qui stimule et entretient la tendance de leur croissance. C’est pourquoi chaque cycle de formation est ancré dans l’analyse de données collectionnées par les enseignants. Il s’agit d’observations personnelles, de protocoles d’entretiens diagnostiques avec plus ou moins d’élèves, de solutions de problèmes posés à des élèves à deux ou trois degrés de l’école, et bien sûr des données du suivi. Un objectif essentiel de ces activités est de comprendre les principes d’une analyse centrée sur les apprentissages à long terme et sur une différentiation structurelle de ces apprentissages. Un autre objectif est de développer une forme d’évaluation au jour le jour et par cycle de leçons qui affine l’évaluation à long terme avec le suivi des élèves. 4.2. Apprendre à regarder, penser et opérer comme le font les enfants Ce qui bloque l’interprétation des données et ainsi aussi la conception de mesures adaptatives est la difficulté pour beaucoup d’enseignants de se mettre ‘dans la peau des enfants’. C’est-à-dire de regarder, de penser et d’opérer comme eux dans les situations journalières de calcul-mathématiques en appliquant leurs notions, représentations et outils de travail. ‘Je ne peux pas regarder dans la tête des enfants’ est la remarque typique

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Figure 4. Quatre indicateurs du niveau de compr´ehension et de solution des e´ l`eves.

des professeurs dans ce contexte. Nous pouvons briser cette barrière en observant les solutions des élèves sous le microscope pour découvrir les différences caractéristiques qui expliquent bon nombre de problèmes de communication au cours des interactions quotidiennes (voir van Hiele, 1973). Cette exploration mène à la reconstruction de quatre indicateurs du niveau de compréhension et de résolution des élèves (Figure 4): la représentation que se fait l’élève du problème posé, le raisonnement qu’il suit et qui guide sa stratégie et ses procédures de solution et le langage qu’il utilise pour présenter, expliquer et défendre son approche et sa solution. Ce type d’activités enthousiasme les professeurs car ils apprennent petit à petit à reconnaître des constellations typiques et à les associer au niveau de compréhension et de solution de leurs propres élèves. Mais surtout – parce que nous reconstruisons ensemble et à partir des données de chacun le ‘paysage mathématique’ que construisent les (groupes d’) élèves d’un même degré, les frontières de ces paysages et les défis à leur horizon (comparer avec Fosnot et Dolk, 2001a; 2001b). Nous remplaçons ainsi, de session en session, les anciennes références par des alternatives et des options testées individuellement dans la pratique et reconsidérées et organisées ensuite par une réflexion collective sur les expériences de classe. 4.3. Chercher la source des problèmes en soi et dans la culture de l’école De telles activités invitent les professeurs à porter leur regard sur leur propre comportement. Elles ouvrent ainsi la porte d’une exploration cri-

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tique de ce qui, en eux, influence ces comportements. Nous pensons aux représentations, suppositions, croyances, opinions et ‘histoires’, construites dans le passé à partir de faits réels et qui figurent depuis comme des vérités. Inspirés par l’approche systémique de Senge (2000), nous utilisons ce qui irrite et frustre les enseignants au cours de leurs activités expérimentales pour mettre à jour ces filtres qu’ils ne contrôlent pratiquement plus et qui modèlent leurs perceptions et leurs interprétations des ‘faits divers’ de leur vie scolaire. Ce type d’activités est d’autant plus délicat qu’il engage les professeurs à mettre en question une grande partie des opinions qui ‘justifient’ leurs routines et à construire eux-mêmes des alternatives fondées sur de nouvelles références et expériences. Orienter ainsi la discussion et la réflexion sur les opinions et routines personnelles met bien sûr la culture et les traditions de l’équipe scolaire et de l’école en question. Les données de l’auto-évaluation peuvent alors orienter les objectifs et les contenus de nouvelles activités de formation plus centrées sur le travail au sein de l’équipe scolaire. 4.4. Amélioration de la pratique par une coopération entre chercheurs, formateurs et enseignants Apprendre à adapter les apprentissages aux acquis des élèves pour favoriser un développement ininterrompu est une entreprise difficile et de longue haleine. Elle implique un tel investissement pour les écoles et exige des compétences si variées qu’il nous semble indispensable de créer un cadre de travail délimité dans l’espace et dans le temps. C’est pourquoi nous préconisons la formation de ‘cercles de qualité’ (Kraemer, 1998). Nous pensons à des réseaux locaux d’enseignants et de membres des services et instituts d’encadrement (chercheurs, professeurs de formation et conseillers pédagogiques) dans lesquels les participants travaillent à leur propre rythme et niveau de compétences. Ils sont tous engagés dans un même processus d’amélioration locale de la pratique qui intègre la recherche d’adaptations concrètes à la formation professionnelle de chaque participant. La pratique de tous les jours est l’objet de recherche et de formation et forme en même temps le cadre naturel d’expérimentation (comparez avec Simon, 1995; Hiebert et Stigler, 1999). Nous avons exploré les quatre principes de formation cyclique sur le tas à petite échelle, dans notre école expérimentale de La Haye et au cours d’un cycle expérimental de formation à Sittard, dans la région du Limbourg. L’évaluation actuelle de ces expériences oriente le développement d’un premier cycle de formation qui sera proposé au cours de 2003. Dans la section suivante, une description de quatre activités expérimentales va concrétiser les principes exposés ci-dessus.

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Figure 5a. Cinq des 18 questions de l’entretien diagnostique du degr´e 2.

5. C YCLE D ’ ACTIVITES DE FORMATION : UN EXEMPLE

Le cycle de formation présenté ci-dessous illustre les principes de notre approche de formation. Nous explicitons dans chaque phase du cycle les objectifs, les activités et les constructions des professeurs qui introduisent de nouvelles questions et les engagent dans un nouveau défi. 5.1. Construire des points de repère qui ont du sens Une manière défiante et efficace d’engager les enseignants consiste à les inviter à s’entretenir avec quelques élèves de leur classe sur quelques questions des épreuves du suivi (Figure 5a). Dans cet exemple l’analyse du protocole de l’entretien avec Patrick et Liane (Figure 5b) permet en premier lieu aux participants d’explorer les différences en reconstruisant les indicateurs de niveau présentés plus haut dans la Figure 4 (représentation mentale de la situation, raisonnement suivi, langage utilisé, stratégie et procédures suivies). De ce travail émergent des opinions et suppositions sur les acquis, les fausses conceptions, les lacunes et les perspectives de progrès chez Patrick et Liane. Les participants apprennent à les expliquer et à les justifier en associant les unes aux autres les informations cachées dans les solutions

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Figure 5b. Extraits du protocole d’un entretien diagnostique au degr´e 4 en janvier/f´evrier.

de Patrick et de Liane. Les discussions dévoilent une certaine cohérence dans leur attitude et leur comportement. Elles permettent aux participants de prendre conscience de la qualité et des limites de leur propre compréhension des mathématiques et du progrès des apprentissages dans ce domaine des nombres et du calcul mental jusqu’à cent. Les différences d’interprétation et d’opinion soulèvent des doutes qui nous permettent d’introduire le profil de compétence de la Figure 6 comme moyen de contrôle de sa propre taxation du niveau de Patrick et de Liane à partir des données empiriques du suivi. 5.2. Explication du graphique L’axe vertical symbolise l’échelle normalisée, les segments verticaux plus ou moins longs les questions de l’entretien. Les numéros gras sont les questions du protocole utilisées ici. La projection de l’origine de chaque

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Figure 6. Profil de comp´etences a` partir des donn´ees empiriques du suivi des e´ l`eves.

segment sur l’échelle donne le niveau de compétence nécessaire pour résoudre 5 questions sur 10 de ce type correctement (paramètre de difficulté). La projection de l’extrémité des segments donne à son tour la compétence exigée pour résoudre correctement 8 questions sur 10 de ce type (paramètre de maîtrise). En traçant les lignes horizontales à partir du niveau actuel de progression de Patrick (52), de Liane (73) et de celui de l’élève moyen (61) les participants distinguent les questions et contenus que maîtrisent ces élèves (tous les segments en dessous de leur niveau), ce qu’ils construisent en ‘ce moment’ (les segments coupés par la ligne horizontale) et ce qui dans cette phase n’a pas encore de sens pour eux (tous les segments au-dessus de leur niveau). Ils peuvent alors contrôler les relations qu’ils ont établies entre les questions et leurs contenus ainsi que leurs suppositions sur les acquis et les lacunes. De tels profils ancrent dans ce sens les repères qualitatifs des participants dans les repères empiriques du suivi. La conclusion de cette première phase est que Patrick fait, comme chacun le pensait, partie du groupe des élèves les plus faibles du degré 2 et que Liane, elle, fait partie des meilleurs élèves de ce degré. Elle maîtrise tous les types de questions, à l’exception des deux derniers (les soustractions du type de la question 18). La question qui émerge de ces premières activités est bien sûr celle du soutien de Patrick et des élèves qui opèrent à son niveau à partir de ce qu’ils comprennent et appliquent correctement ‘en ce moment’ et dans les perspectives du programme scolaire de leur degré.

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Figure 7. Profil hypoth´etique de Patrick.

5.3. Mieux comprendre la force mathématique et les limites des élèves ‘faibles’ en reconstruisant leur tendance de développement En analysant les protocoles, les participants ont exploré les concepts, idées, représentations, raisonnements, stratégies, procédures de Liane et de Patrick. Ils ont délimité avec eux grossièrement le champ de différentiation au degré 2. Il s’agit maintenant de mieux comprendre les possibilités et les limites des élèves comme Patrick en reconstruisant la tendance de développement à leur âge à partir des données d’entretien avec d’autres élèves qui opèrent à des niveaux intermédiaires (entre celui de Patrick et celui de Liane). Les participants établissent dans ce but des profils hypothétiques de compétences, tels que celui de la Figure 7. Ils lient les données de ces profils les unes aux autres et explorent ainsi les changements plus ou moins subtils au niveau de la conception des nombres, de l’organisation de ces nombres et des opérations avec ces nombres. En travaillant ainsi, les participants reconstruisent ‘pierre par pierre’ le paysage mathématique qu’inventent et organisent les élèves, réinventent et affinent les outils mathématiques qu’ils fabriquent et explorent leur utilisation dans les situations courantes des leçons. Une telle reconstruction de la tendance de développement provoque en général de nombreuses remarques sur les différences entre les élèves telles que les professeurs les vivent dans leur classe. Les anecdotes qu’ils racontent à propos des problèmes que soulèvent ces différences portent alors l’attention sur les routines qu’ils ont développées pour tenter de les résoudre. Elles ouvrent ainsi la porte de l’introspection de la phase suivante.

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5.4. Identifier en soi la source de problèmes persistants C’est l’ambiance ‘ouverte’ de travail qui invite les participants à prendre le risque d’observer ‘en eux’ ce qui pourrait expliquer les problèmes qui ‘reviennent’ constamment et qui les mènent toujours dans les mêmes impasses. Nous avons ‘provoqué’ l’exemple d’introspection de la Figure 8 à partir d’une leçon expérimentale donnée par chaque participant dans sa propre classe. L’objectif était triple: a) inventorier les stratégies et procédures de multiplication plus ou moins formelles utilisées par les élèves dans leur résolution du problème posé; b) organiser ces stratégies et procédures dans une séquence de progression à partir de l’expérience de la session précédente; c) décrire un problème typique et persistant lié à la gestion des interactions au cours de telles leçons et découvrir ce qui ‘en soi-même’ pourrait expliquer la persistance de ce problème. Yvonne est une enseignante très consciencieuse qui s’est engagée totalement au cours de ce cycle. Son entretien ‘avec elle-même’ a beaucoup frappé ses collègues, car ils ont reconnu dans son comportement et sa réflexion leurs propres habitudes, rationalisations et frustrations. Ce que nous avons retenu de cette réflexion est la difficulté de prendre le temps qu’il faut pour comprendre ce qu’expose un enfant et de contrôler cette compréhension. Mais aussi le besoin d’apprendre à apprécier la valeur de chaque construction en en comprenant sens dans le processus long de développement en question. Dans ce sens, de telles introspections stimulent la curiosité des professeurs et les engagent à développer les références, les connaissances, les compétences et les formes de travail dont ils ont besoin pour exploiter au maximum les constructions de chaque élève. Les activités de la dernière phase du cycle donné en exemple sont centrées sur cet objectif. 5.5. Apprendre à adapter en jetant un pont entre Patrick et Liane Dans cette dernière session du cycle nous faisons un lien entre les remarques de la session précédente et un fragment du protocole de l’entretien avec Patrick réservé dans ce but. Patrick est en effet un de ces élèves dont parle Yvonne et qui provoquent (toujours) les mêmes réactions du professeur et du reste de la classe. Ses explications prennent en général beaucoup de temps puisqu’il utilise surtout des procédures de comptage difficiles à comprendre par son utilisation très personnelle des doigts comme modèle de calcul. Le dernier travail collectif de la Figure 9 consiste à jeter un pont entre l’approche ‘bizarre’ de Patrick et la stratégie de calcul mental de Liane qu’attendent les professeurs à ce degré: soustraire en complétant. Les questions posées permettent aux participants de réfléchir sur les principes didactiques qu’ils appliquent journalièrement, la trajectoire

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Figure 8. Introspection d’Yvonne avec elle-même.

Figure 9. Qu’est-ce qui lie Patrick a` Liane?

d’apprentissage qu’ils suivent avec leurs manuels et les matériaux qu’ils utilisent pour orienter les élèves et soutenir le processus de développement et de formalisation graduelle des procédures de calcul mental. Les participants reconstruisent en fait le chemin qui mène à l’idée d’utiliser une ligne ‘vide’ pour modéliser différentes stratégies et procédures de soustraction à partir du sens que l’on peut donner à ces soustractions (Figure 10). C’est ainsi qu’ils apprennent à apprécier les ‘inventions’ les

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Figure 10. Modelage de deux strat´egies et proc´edures de solution a` partir de deux interpr´etations du probl`eme des maisons.

plus rudimentaires et à les exploiter dans la ligne de pensée des enfants et des auteurs de leurs manuels. Les adaptations qui émergent de telles activités sont ‘testées’ dans la pratique. En rapportant leurs expériences au début d’un nouveau cycle d’activités, les participants peuvent reconsidérer les hypothèses et opinions de départ et s’engager dans de nouvelles explorations pour affiner les repères, les connaissances et les principes développés jusque là. C’est par de tels cycles d’activités que les participants développent petit à petit un nouveau cadre de référence et une panoplie plus ou moins large d’outils qui donnent confiance et leur font accepter les risques de défier chaque élève à son niveau.

6. R ÉFLEXION FINALE

L’exemple d’activités illustre les deux principes clés de notre approche de formation: – utiliser un suivi des élèves pour ancrer les observations diagnostiques des participants dans des données empiriques et pour orienter et régulariser l’adaptation des objectifs et des contenus intermédiaires de la progression; – centrer les activités de formation sur a) une meilleure compréhension et appréciation des constructions et du progrès des élèves, b) explorer les facteurs en soi qui entretiennent les routines et bloquent ainsi le changement et c) développer et intégrer progressivement de nouvelles

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attitudes, opinions et modes de travail de plus en plus adaptés aux besoins et au potentiel de croissance des enfants. C’est dans ce sens que nous considérons le suivi des élèves comme un instrument indispensable de pilotage de l’innovation et de l’amélioraton de la pratique. Nos experiences de formation soulèvent trois points de réflexion à ce propos. 6.1. Intégrer les formes d’évaluation de la pratique à partir d’une réflexion systémique sur la gestion des apprentissages L’impression que donne notre suivi dès son application dans les écoles est que c’est un instrument fiable, facile à intégrer dans l’organisation de la classe (des écoles) et qui confirme les évaluations des professeurs au jour le jour. Cette impression explique l’enthousiasme des professeurs tant qu’ils utilisent ce suivi ‘á côté’ de l’évaluation, telle qu’ils la pratiquent par tradition (observations, entretiens et corrections journalières, épreuves périodiques des livres utilisés, etc.). Le suivi fonctionne en effet dans cette phase comme un ‘révélateur externe’ des différences que les professeurs ‘connaissent’ déjà par leur ‘évaluation interne’. Les données des épreuves n’influencent pas les décisions d’adaption, mais justifient plutôt les mesures prises à partir de l’évaluation de tous les jours. Dans ce sens, le problème clé de l’utilisation d’un suivi comme instrument de pilotage est l’intégration de toutes les formes d’évaluation en tenant compte des rôles et des besoins des professeurs dans les différents contextes de la gestion des apprentissages ainsi que de la complémentarité des formes et outils d’évaluation. Quelles données allons-nous utiliser pour nos rapports mensuels ou trimestriel aux parents et pour adapter notre programme? Celles des épreuves périodiques du livre ou celles du suivi? Cette question pratique des professeurs de notre école expérimentale nous a signalé ce besoin d’intégration. Elle en soulèvent mille autres que l’équipe scolaire ne peut traiter avec bon sens que par une réflexion systémique sur les aspects clés de l’enseignement différentié, à partir des problèmes auxquels se heurtent les professeurs dans leur pratique et de leurs besoins les plus urgents. Un suivi des élèves est, certes, un instrument indispensable pour gérer les apprentissages à longs termes. Mais les professeurs ne transforment pas automatiquement leur pratique d’évaluation et de gestion des apprentissages par son utilisation. Créer un espace et un temps de formation est selon nous la condition sine qua non pour réaliser cet objectif en engageant les professeurs dans une formation continue centrée sur le développement personnel de chaque participant.

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6.2. Mieux comprendre les exigences d’un enseignement qui défie en réfléchissant sur ses propres apprentissages C’est à partir de ce constat que nous engageons les professeurs dans un processus de réflexion sur et de recherche dans leur pratique et d’investir par-là dans leur propre formation. L’idée centrale est d’apprendre à problématiser la question complexe de l’adaption des objectifs et contenus du programme et de développer soi-même, en dialogue avec ses collègues, un cadre de référence, une façon de penser et des méthodes et outils de travail applicables dans les situations les plus courantes de la pratique. Par notre approche active et systémique de formation (Senge et al., 2000) nous défions dans ce sens les professeurs à rationaliser leur pratique comme ils sont sensés de défier leurs élèves à mathématiser le monde dans lequel ils vivent (Freudenthal, 1968). La formation reflète autrement dit les exigences d’apprentissages défiants et qui stimulent les acteurs à prendre leur propre part de responsabilité. Elle permet ainsi aux professeurs de mieux comprendre leurs responsabilités essentielles envers leurs élèves. 6.3. Pas de standards sans données empiriques La politique actuelle de décentralisation de l’enseignement et de régulation des innovations incite de plus en plus de pays à formuler des standards que devraient réaliser le plus grand nombre d’élèves dans le plus grand nombre d’écoles. Notre comparaison des résultats scolaires (dans le cadre des sondages périodiques du niveau de l’enseignement) avec les standards néerlandais formulés par les enseignants, professeurs de formation et consultants scolaires signalent que ceux-ci sousestiment systématiquement le niveau de difficulté des objectifs que 70 à 75%% des élèves devrait réaliser selon eux à la fin du primaire. Ils estiment par contre plus ou moins correctement les standards qu’ils formulent pour la minorité d’élèves très en retard et très en avance. Cette constatation jette des doutes sur le sens de formuler des standards à partir des opinions des responsables de l’enseignement et des experts des services d’encadrement (ou sur la base d’une consultation plus large) sans références empiriques sur la progression réelle des enfants au cours de leur scolarité obligatoire. Ceci nous pousse à préconiser une discussion sur les orientations d’un enseignement adapté aux exigences de la vie actuelle à partir d’une vue d’ensemble sur ce qu’apprennent et appliquent les élèves actuellement entre 4 et 16 ans à différents niveaux de compréhension et d’aptitudes. Nous manquons de tels repères pour justifier pleinement la différentiation des objectifs et contenus au cours du cycle primaire et pour orienter les élèves dans la voie qui correspond le mieux à leurs affinités et leurs talents.

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N OTES 1. Cet article est une adaptation de la conférence plénière de l’auteur à la 54ème Rencontre de la Commission Internationale pour l’Etude et l’Amélioration de l’Enseignement Mathématique à Vilanova i la Geltrú (Catalogne, Espagne), 13–19 juillet 2002.

R EFERENCES Eggen, T.J.H.M. et Sanders, P.F. (eds.): 1993, Psychometrie in de Praktijk (Psychométrie dans la Pratique), Cito Instituut voor Toetsontwikkeling, Arnhem. Fosnot, C.T. et Dolk, M.: 2001a, Young Mathematicians at Work. Constructing Number Sense, Addition, and Subtraction, Heinemann, Portsmouth, NH. Fosnot, C.T. et Dolk, M.: 2001b, Young Mathematicians at Work. Constructing Multiplication, and Division, Heinemann, Portsmouth, NH. Freudenthal, H.: 1968, ‘Why to teach mathematics so as to be useful? Educational Studies in Mathematics 1(1), 3–8. Gauthier, P.-L.: 2002, ‘Du bon usage de l’évaluation’, dans Revue Internationale d’Éducation. L’évaluation des systèmes éducatifs, N◦ 26, juin. Sèvre: Centre international d’études pédagogiques, pp. 15–18. Giménez, J., Keitel, C., Hahn, C., Luelmo, J., Paola, D., Santos, L. (Comité du programme de la CIEAEM54): 2002, Documento de debate: Un reto para la educación matématica: reconciliar lo común con la diversidad. http://www.upc.es/info/cieaem54/cieaem-cas/2anunci.htm Janssen, J.: 2002, Leerlingvolgsysteem 2002 (Suivi des élèves 2002), Citogroep, Arnhem. Janssen, J., Schoot, F. van der, Hemker, B. et Verlhelst, N.: 1999, Balans van het RekenWiskundeonderwijs Einde Basisschool 3. Uitkomsten van de Derde Peiling in 1997 (Balance de l’Enseignement des Mathématiques à la Fin de l’école Primaire. Résultats du 3ème sondage de 1997), Citogroep, Arnhem. Janssen, J. et Kraemer, J.M.: 1995 et 1997, Leerlingvolgsysteem. Rekengids 1 & 2. (Suivi des élèves. Guide de mathématiques), Citogroep, Arnhem. Hiebert et Stigler: 1999, The Teaching Gap, Free press, New York. Hiele, P.M. van: 1974, Begrip en Inzicht, Muusses, Purmerend. Kraemer, J.M.: 1998, Kwaliteitscirkels in Haaglanden. Interne Nota. (Cercles de qualité autour de La Haye. Notice interne), Citogroep, Arnhem. Kraemer, J.M.: 2001, ‘Desafíos de la enseñanza de las matemáticas en la escuela primaria holandesa’, dans G. Giménez (coord.), P. Abrantes, et L. Bazzini, Comisión CIEAEM, Grup 100, C. Keitel, J.M. Kraemer et S. Romero, Matemáticas en Europa: Diversas perspectivas, Graó., pp. 51–71. Kraemer, J.M.: 2002, Leerlingvolgsysteem. Hulpboeken voor Groep 3 tot en met 5. (Suivi des élèves. Livres de soin pour les Degrés 1, 2 et 3), Citogroep, Arnhem. Kraemer, J.M. (sous presse), Leerlingvolgsysteem. Hulpboeken voor Groep 3 tot en met 5. (Suivi des élèves. Livres de soin pour le Degrés 4), Citogroep, Arnhem. McIntosh, A., Reys, B.J. et Reys, R.E.: 1992, ‘A proposed framework for examining basic number sense’, For the Learning of Mathematics 12(3), 2–8. Michel, A.: 2000, ‘Évaluer pour piloter’, dans Revue Internationale d’Éducation. L’évaluation des Systémes éducatifs, N◦ 26, juin. Sèvre: Centre international d’études pédagogiques, pp. 19–29.

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Schoot, F. van der et Dam, P. van: 2000, Professionnalisation de l’évaluation aux PaysBas. Revue Internationale d’Éducation. L’évaluation des systémes éducatifs, N◦ 26, juin. Sèvre: Centre international d’études pédagogiques, pp. 73–81. Senge, P., Cambron-McCabe, N., Bryan Smith, T.L., Dutton, J. et Kleiner, A.: 2000, Schools that learn, A fifth Discipline Fieldbook for Educators, Parents, and Everyone Who cares about Education, Doubleday, New York. Simon, M.: 1995, ‘Reconstructing mathematics pedagogy from a constructivist perspective’, Journal for Research in Mathematics Education 26(2), 114–145.

Centre National d’Évaluation en Éducation, Citogroep, Pays-Bas

RONNIE KARSENTY

WHAT DO ADULTS REMEMBER FROM THEIR HIGH SCHOOL MATHEMATICS? THE CASE OF LINEAR FUNCTIONS

ABSTRACT. A qualitative study was designed to investigate adults’ long-term memory of mathematics learned in high school. Twenty-four men and women, aged 30 to 45, were requested to recall mathematical concepts and procedures during individual interviews. This article reports findings regarding the subjects’ attempts to draw graphs of simple linear functions. In general, these findings support the idea that retaining high school mathematical content strongly depends on the number, level, and total length of mathematics courses taken by the student. Diverse responses to the task of drawing a graph of a linear function such as y=2x, were documented and categorized. In many of these responses, the basic mathematical communal notion of linear graphing was replaced with personal on-the-spot constructing of ideas. Detailed analysis of three cases is presented, based on recall theories that explain the mechanism of recalling in terms of reconstruction vs. reproduction. KEY WORDS: adults, functions, graphs, long term memory, qualitative methods, recall theories

1. T HE LONG - TERM MAINTENANCE OF KNOWLEDGE LEARNED IN SCHOOL :

A ROAD ( ALMOST ) NOT TAKEN

The educational community is a rather multifaceted one. Different sectors in this community – parents, teachers, researchers and other members – do not always see eye to eye, to say the least. However, if there is something that educators do share, I suppose that it is the basic aspiration that makes us send our children to twelve years of schooling: We all want our children to be ‘educated’, to know things about the world, to broaden their horizons. Bearing in mind this naive but probably most agreed-upon goal, it is quite surprising that of the immense existing body of school research, only a minute portion is dedicated to questions of knowledge maintenance after school. Although the scarcity of research in this field was referred to more A short version of this paper was presented at the 2002 PME 26 conference in Norwich, UK: Karsenty, R. and Vinner, S.: 2002, ‘Functions, many years after school: What do adults remember?’, in A.D. Cockburn and E. Nardi (eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, Vol.3, University of East Anglia, Norwich, pp. 185–192.

Educational Studies in Mathematics 51: 117–144, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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than twenty years ago by memory researchers Bahrick (1979) and Neisser (1978), their call for innovative ecological memory studies was, in regard to this specific issue of knowledge learned in school, not widely answered. The following citation from Bahrick (1979) expresses his dissatisfaction with the state of affairs at the time, and it is still quite relevant today: It is disappointing that nearly 100 years of research have not yielded much progress toward specification of the conditions under which information, once acquired, can be maintained indefinitely. [. . .] Much of the information acquired in classrooms is lost after the final examinations are taken, but beyond the general advice to practice and rehearse frequently, we have little to offer those who wish to minimize or prevent such losses. (Bahrick, 1979: 297)

Nevertheless, some research on long-term memory for knowledge learned in school does exist. Semb and Ellis (1992, cited in Semb et al., 1993), who reviewed studies concerning this issue, pointed out that some of these studies were difficult to trace, due to the fact that they appeared in disciplinespecific journals. Indeed, if the small aggregate of relevant research is to be sorted by disciplines, we might find that for certain school subjects there are few studies, at most, which address the issue of how much is remembered of those subjects in the after-school years. In regard to mathematics, a unique work is that of Bahrick and Hall (1991). This large-scale quantitative study was designed to identify variables that affect losses in recall of high school algebra and geometry contents, throughout the life span. A major finding of this work was: When the acquisition period extends over several years, during which the original content is relearned and used in additional mathematics courses, the performance level at the end of the training is retained for more than 50 years, even for participants who report no significant additional rehearsal during this long period. In contrast, those whose acquisition period is limited to a single year perform at near-chance levels. . . (Bahrick and Hall, 1991: 30)

Bahrick and Hall also found that grades in courses, as well as standardized test scores, have minor influences on the rate of loss, although they reflect differences in the original degree of knowledge. The rate of loss is most affected, as emerges from the citation above, by variables concerned with the amount and distribution of practice. This conclusion is in agreement with findings from several memory studies in other disciplines of knowledge (Semb et al., 1993), and we will return to it later on. However, Bahrick and Hall’s work does not tackle the following question directly: what do adults remember from their past mathematical studies? The performance of Bahrick and Hall’s research participants (about 1700 in number) was measured by psychometric means (i.e., correct/incorrect answers). For obvious reasons, a statistical study of such a large scale does not usually involve cognitive analysis of answers to open-ended questions.

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Since cognitive analysis could potentially reveal phenomena that remain unnoticed by psychometric analysis (Karsenty and Vinner, 1996; Cooper and Dunne, 2000), it seems that a qualitative study, scrutinizing adults’ responses to mathematical tasks, could add to the general picture in the issue of maintenance of high school mathematical content. Thus, the research reported herein concentrated on a small group of adults, in order to allow an in-depth analysis. The analysis was based in part on theories of recall, which will be described in the next section. 2. T HEORIES OF RECALL : A BRIEF REMINDER

The ways in which people recall different contents have long been an attractive subject for researchers. However, almost all of the traditional memory studies were held in research laboratories, and hence referred mostly to contents that could be acquired in laboratory settings (e.g., words, scripts, lists of various kinds). The well-acknowledged classic work of Bartlett (1932) laid the foundations of this field. In Bartlett’s famous research, subjects were requested to recall a story they had read, within several time intervals. After meticulous examination of the discrepancies between different versions of the story, and based on other experiments, Bartlett suggested that recalling is a mechanism of reconstruction rather than reproduction. These terms will now be explained briefly. Reproduction means that specific details from the past are coded in memory, in such a way that enables the eliciting of ‘copies’, however pale or dim, in the present. Prior to Bartlett’s publication, most theories of recall focused on the idea of reproduction, as can be recognized from various titles given to these theories: ‘copy theories’, ‘trace theories’ and ‘reappearance hypothesis’ (see Neisser, 1967). Bartlett rejected these theories, suggesting that recall is performed through a process of reconstruction: This means that attempts to elicit past experiences tend to yield an altered version, that might be different from the original one in a substantial manner. For example, when requested to repeat a story, people are likely to produce an interpretation, though they may be unaware of doing so. Thus, some details might be omitted, others emphasized or even added. According to Bartlett (1932), reconstruction is controlled by schemas, which he defined as active organizers of past experiences. More recent memory researchers, such as Neisser (1984), basically agree with Bartlett’s ideas, although they suggest some refinements and modifications. Brewer and Nakamura (1984), for instance, talk about partial reconstructive recall. Yet, the context of many discussions about the nature of the recalling mechanism, is what Brewer (1986) called personal memory, or, to use Tulving’s (1972)

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original distinction, episodic memory1 . These terms refer to the recall of events and occurrences personally experienced by an individual. Theories of recall are less frequently applied to depersonalized knowledge, much less to knowledge learned in school. In this article I would like to offer a glimpse into adults’ attempts to recall mathematical material, in a way that suggests that a mechanism of reconstruction might take place. Moreover, I intend to show that such a reconstruction, idiosyncratic as it might be, sometimes follows a certain inner logic that is used in the absence of accessible relevant details.

3. T HE STUDY: C HARACTERISTICS AND METHODS

3.1. The type of research conducted The research presented here is defined, according to Stake’s classification (1994, 1995), as a collective case study. This category of qualitative research refers to studies in which a certain number of cases are thoroughly examined in order to highlight a particular issue. In contrast to an intrinsic case study, in which the focus of interest is on the specific case uniquely, a collective case study is usually defined as instrumental, that is, the analyses of cases are meant to serve as a vehicle for enhancing a more general understanding in regard to some phenomena or theory. However, as Stake (1994) notes, the primary purpose of case studies, even when they are collective and instrumental, should be to gain a better understanding of the cases themselves. This study was therefore designed in light of two complementary objectives: to learn as much as possible from each single case, but also to obtain a more general picture in regard to all cases. This dual intention affected the case selection procedure, as will be described below. 3.2. Target population and case selection The study took place in Israel, and was restricted to Israeli high school graduates who had passed the Israeli Matriculation Exam in mathematics, which is an external, nation-wide test, taken at the end of high school. The first decision that had to be made in regard to subjects’ selection was the sector from which to choose candidates: Clearly, an attempt to represent the whole range of high school graduates would have been unfeasible, since the number of cases was to be relatively small. Therefore it was decided to focus on adults with post-secondary education. The implied assumption was that the voices of subjects holding college or university

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degrees might define ‘the upper bound’ of people’s recollections in regard to mathematics. The second methodological decision concerned the issue of accessibility. The main research method was an extensive interview. Such a procedure is, on the subject’s part, not only time consuming, but also – and this might be even more crucial – emotionally demanding. Hence it could not be expected that randomly chosen adults would cooperate with a researcher unknown to them. It became clear that some degree of personal acquaintance between the researcher and the subjects was a prior condition to the study’s implementation. After examining different social circles in which I had some involvement, I decided to select subjects from the population of a small village near a city where I was living at the time. This village was founded twenty years ago, and the adult residents – approximately 200 in number – came from all over the country and had attended different high schools in their youth. Members of this community can be seen as representing the higher-educated sector of the society in Israel. Restricting the study to volunteers between the ages of 30–452 who were not mathematics graduates or mathematics teachers, led to a group of 105 candidates. Then, in light of the dual objective mentioned above, personal data about the candidates were gathered, in order to maximize diversity when choosing subjects. The goal was to achieve, within the limited sector I focused on, a wide spectrum of cases that would consequently enable a broader view, based on investigating potentially different past mathematics learners. Three factors were considered: gender, level of mathematics taken in high school3 and current profession4 . After mapping these characteristics in a sampling table, a subject was chosen randomly from each non-empty cell. The final number of subjects was 24. Of these, 12 were men and 12 women, 12 took mathematics in the low-level track and 12 in the intermediate or high level tracks (see note 3). All subjects had post-secondary education, mostly from colleges or universities, and they were all engaged in professional careers in areas such as law, medicine, psychology, art, business, high school teaching and other. 3.3. Research methods Each subject participated in an individual session, which lasted between 2 to 3 hours. A lion’s share of this time was dedicated to a semi-structured interview, consisting of two parts. The first part was dedicated to affective aspects, later analyzed within a framework I called “the personal mathematical profile”. In this article I will not refer to this affective part of the research. Some results can be found in Karsenty and Vinner (2000). Detailed results will be published elsewhere.

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In the second part of the interview, subjects were asked to solve mathematical tasks involving basic concepts and procedures. With a list of nine questions as a starting point, the conversation was constructed spontaneously according to the flow of ideas expressed by the subject. If interesting themes emerged, usually further unplanned questions were posed in order to follow the subject’s line of thought. Moreover, it is important to emphasize that the interview was interactive, as the interviewer occasionally responded to the subject’s answers. Responses were either in the form of feedback when necessary (e.g. if it became clear that the subject would not continue without it), or brief reminders, instruction, or hints in case the subject was ‘stuck’ in the process of recall. The intention was to investigate if and how a certain memory could be elicited through a certain trigger. The interviews were recorded and later transcribed. Of the data collected in the research, I will report here on results concerning the subjects’ attempts to draw graphs of simple linear functions, as documented during interviews. These results are presented in the next section.

4. R ECALLING GRAPHICAL DESCRIPTIONS OF SIMPLE LINEAR FUNCTIONS

4.1. Two levels of reporting subjects’ reactions to the task Subjects were requested to draw a graphical representation of a simple linear function. In most cases the function was y=x. However, in some cases the function was different, for instance in the case of Ilan5 , who mentioned the function y=2x during a previous answer, and thus was asked to draw it, in the natural flow of the interview. Usually, the discussion with each subject around the issue of drawing the given function did not end quickly. In fact, of all the tasks presented during interviews, this one produced some of the richest and most comprehensive data. In many cases, after the subject gave a spontaneous answer, and was asked to explain this answer, the discussion evolved in one (or a combination) of the following ways. Either the subject thought some more and changed or refined the original answer, or the subject justified the answer in a way that evoked further questions in regard to other functions. In addition, as the discussions involved mild interventions, in the manner explained above (see section 3.3), this process yielded some further interesting responses. In light of the extensive data, introducing results is a complicated and problematic issue, to be considered carefully. Since reporting different elements in each of the 24 discussions was unfeasible, it was decided to report the results in

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Figure 1. Shaul’s sketch for the function y=2x.

two levels. First, the spontaneous answers will be introduced, grouped into categories. This level of report gives a general picture about what people remember on the surface. Then, three cases will be reported and analyzed in detail, including the evolution of ideas during discussions. This level of report is intended to exemplify processes of reconstructing memory. 4.2. Research results (1): Subjects’ spontaneous responses to the task of graphing a linear function The initial attempts of all subjects to draw the function y=x (or some other linear function) were grouped into six categories, described below. Category I. Sketching a correct graph by marking two or three points in a coordinate system and connecting them with a straight line. Category II. Sketching a straight line that reflects a misinterpretation of the relationship between x and y. Example: Shaul (male), a 45-year-old sound technician who studied mathematics in the high level track, sketched the graph of y=2x shown in Figure 1. His reasoning was: S:

If y equals two x, so each segment in y equals two segments of x. [. . .] So it’s about here, right? [Marks points at (2,1), (4,2) and draws a straight line through them].

Shaul translates y=2x as “having two x’s for each y”. This translation re-

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Figure 2. Gadi’s sketch for the function y=2x+1.

flects a well-known proportional misconception, usually referred to in the literature from the opposite direction, i.e., mistranslating word sentences describing relations into symbols, as in the famous student-professor problem (Rosnick and Clement, 1980; Rosnick, 1981; Kaput and Sims-Knight, 1983; Philipp, 1992). However, if we are to look beyond this misconception, Shaul’s answer communicates the general idea of using two points as means of drawing a straight graph in a Cartesian system. Category III. Sketching an incorrect graph based on a holistic estimation of the behavior of the function. Example: Gadi (male), a 41-year-old architect, who studied mathematics in the intermediate level track, sketched the graph of y=2x+1 shown in Figure 2. His reasoning was: G:

y is at least twice as big as x, so on every move of 1 here [on the x-axis], I need here [on the y-axis] 2 and more.

Gadi does not plug numbers in the given function, but rather analyzes the relationship between x and y, and draws the line accordingly. A further question posed to Gadi, requesting him to draw y=2x+3, yielded another straight line through the origin, with a steeper slope. Gadi explained: “So it’s a little more. Not much. What counts is the ratio of 2”. Two motives are revealed by Gadi’s reasoning. First, Gadi appears to be confident that these two functions are represented by straight lines. It should be noted that later in the interview, when asked about the function y=x2 , Gadi sketched the correct parabola – again without plugging in numbers, but as a holistic pic-

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ture preserved in memory. This link between the symbolic and the graphic representations can be regarded as an example of what Skemp (1976) called relational understanding. Indeed, Skemp suggested that such understanding of mathematical content has a positive impact on maintaining this content for long periods of time. Second, Gadi recalls both straight graphs (y=2x+1, y=2x+3) as starting from the origin. He regards the ratio m as the dominant aspect in the linear expression y=mx+n, while n appears to have less effect, yet they both determine the slope. This idiosyncratic idea seems to have taken over the original learning. Category IV.

Marking only one point in a coordinate system.

Example: Amira (female), 31, a museum director who studied mathematics in the intermediate level track, was requested to draw the graph of the function y=x. In response, she sketched a Cartesian axes system and marked the point (1,1). Her reasoning was: A:

You said that x is equal to y, and if this is x and this is y, and these are the positive points, and this is 1 and this is 1, so let’s say I did it in the middle.

Amira’s response suggests that the request to draw the graph of y=x is interpreted as “solving”, i.e. finding a point in the Cartesian plane where y is indeed equal to x. The point marked is an arbitrary representative of solutions to the equation y=x. Category V. Drawing a graph by allocating segments on the x-axis and the y-axis and connecting the two endpoints. Example: Tamar, (female), 42, a high school art history teacher, who studied mathematics in the low level track, sketched the graph of y=x shown in Figure 3. Her reasoning was: T:

Well, anyway, there’s got to be something equal here.

Tamar’s explanation is based on an attempt to make sense of the symbol phrase “y=x” in the context of an axes system, a context that she recalls. Tamar develops an idea that seems to be an improvisation created on the spot. However, we shall see later that this idea is rather persistent. The full discussion with Tamar will be presented in section 4.3.

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Figure 3. Tamar’s sketch for the function y=x.

Figure 4. Yafa’s sketch for the function y=x.

Category VI. Describing the function through equality between shapes or line segments. Example (a): Yafa, (female), 38, a high school humanities teacher, who studied mathematics in the high level track, sketched the graph of y=x shown in Figure 4. Example (b): Dov, (male), 37, a government official, who studied mathematics in the low level track, sketched the graph of y=x shown in Figure 5. These two examples reflect a switch from ideas shared by the mathematical community to idiosyncratic reasoning. The distribution of subjects within the categories I-VI (and a seventh ‘no response’ category) is shown in Table I. Observing this table, a connection can be noticed between high school mathematics level tracks and

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Figure 5. Dov’s sketch for the function y=x.

TABLE I Distribution of subjects within the categories formed for drawing a graph of a linear function (N = 24) Category

No. of subjects assigned to this category, distributed by level of math taken in high school High Medium Low Total in level: level: level: this category:

I. Correct graph. II. Straight line reflecting a misinterpretation of the relationship between x and y. III. Incorrect graph based on holistic estimations. IV. Marking only one point on an axes system. V. Allocating segments on the x-axis and the y-axis and connecting the two endpoints. VI. Equality between shapes or line segments. VII. No response.

5

1

1

7

1

–

4

5

–

2

–

2

–

1

2

3

–

–

3

3

1 –

1 –

1 1

3 1

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Figure 6. Tamar’s sketch, drawn while explaining what is a function.

the category distribution: Most of the subjects classified within the three top categories (9 of 14), which in general can be characterized as expressing some degree of acquaintance with representations of linear functions in a Cartesian system, are subjects who participated in the high and intermediate tracks in high school. Most of the subjects classified in the other four categories, are low-level graduates (7 of 10). In these categories, the basic notion of linear graphing is replaced with personal on-the-spot constructing of ideas. In the next section, some of these ideas will be scrutinized. 4.3. Research results (2): Three cases of reconstructing memory 4.3.1. The case of Tamar As mentioned above in section 4.2, Tamar is a 42-year-old high school teacher. She teaches art history, a subject in which she has a Bachelor’s degree. Tamar studied mathematics in the low level track, and her final matriculation grade was 90. Earlier in the interview, when Tamar was asked what a function is, she said the following, while sketching the drawing presented in Figure 6. T:

There is a horizontal axis and a vertical axis, and there are points. You connect the points and you get a function.

Note that Tamar is using x and y coordinates in her drawing. However, right after that drawing was done, when Tamar was requested to draw a graph for the function y=x, she abandoned this use of coordinates and drew the sketch presented earlier in Figure 3. As explained in Category

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Figure 7. Tamar’s sketch for the function y=2x.

V above, she allocated equal segments of one unit on both axes. She then joined the two endpoints by a line. In light of this description, Tamar was asked to draw the function y=2x. As can be seen, Tamar persisted with the same logic, drawing the graph of y=2x presented in Figure 7. This time the segment allocated on the x-axis (which is the vertical axis in the drawing) is twice as long as the one allocated on the y-axis, thus reflecting a proportional misconception held also by other subjects (see category II in section 4.2 above). After Tamar sketched these drawings for y=x and y=2x, the following conversation evolved: T: Int: T: Int: T: Int:

It’s something like that, I don’t know. Have you any way to check this? No, I don’t know what this means. You said that there are points, you connect the points and you get a function. Yes. [. . .] So, show me for instance a point here [referring to the graph of y=x] that keeps this rule, y=x.

Tamar responds by marking a point in the middle of the ‘hypotenuse’ created by the graph and the axes. The conversation continues:

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Figure 8. Tamar adds a parallel line to the original sketch of y=x.

Int: T: Int:

Why? Because the distance here is equal to the distance here [refers to the two halves of the hypotenuse]. I see. Can you give another one, another point on the graph that shows that its x is equal to its y?

At this point Tamar adds a second line, parallel to the first one, as shown in Figure 8. She explains: T: Int: T:

What is a point? A place on these axes where this is equal to this? It could be anything. Must it be on the axes? I don’t know, yea, yea, sure. Along all the axes, this equals this, this equals this. . .

[Tamar adds more parallel lines, as shown in Figure 9]. Int: T:

So the function is what, this collection of lines, one of these lines? Maybe the function is this.

[Tamar bisects all the lines with one line, as shown in Figure 10] Int: T: Int: T:

Ah. Why? Because everywhere this distance equals this distance. I see. And what would you do here, with y=2x? Ah, now I’m recalling.

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Figure 9. Tamar adds more lines to the sketch of y=x.

Figure 10. Tamar bisects the lines.

Now Tamar repeats the same process with y=2x, only this time the final line does not bisect the parallel lines, but divides them into thirds and two thirds (Figure 11). In order to examine this consistent line of thought, another question is posed to Tamar:

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Figure 11. Tamar modifies the graph of y=2x.

Int: T: Int: T:

Ok, now I’d like to know about the point (2,3). Where is this point? Here there’s 1,2,3 [marks notches on a vertical axis] and here there’s 1,2 [marks notches on a horizontal axis, see Figure 12]. And where is the point (2,3)? Here, in the middle? [refers to the midpoint of the line that joins the two points].

The conversation with Tamar brings forward two colliding ideas. On the one hand, the notion of function appears to be preserved in Tamar’s memory, although as an image rather than a definition (Vinner, 1983). Her sketch, drawn while trying to recall this concept, clearly shows that the concept image includes Cartesian coordinates (see Figure 6). On the other hand, however, Tamar constructs a new schema for the process of graphing functions. This schema is embedded in some current common sense, but interestingly, Tamar refers to it using the phrase “now I’m recalling”. In Bartlett’s terms (1932, see section 2 above), this is an example of reconstruction of memory. The new reconstructed schema gains priority over the original recalled image. Tamar is remarkably persistent in following her own idea; at the end of the cited conversation, she abandons the coordinates even as descriptors of a single point, in favor of applying her new method to describe the point (2,3).

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Figure 12. Tamar draws the point (2,3).

Figure 13. Nir’s first attempt to draw the function y=2x.

4.3.2. The case of Nir Nir, 43, studied professional photography in a college of technology, but is currently working as a furniture designer and a carpenter. In high school, he studied mathematics in the intermediate level track. Nir’s first attempt to draw the function y=2x was classified within Category VI, and is shown in Figure 13. While drawing, Nir said: N:

Two triangles that, taken together, give one big triangle. This is how I see it today, and I know it’s not correct in the mathematical way of thinking.

At this point, a small intervention was made in attempt to trigger Nir’s recalling process, and Nir responded to it immediately, as can be seen in the following paragraph:

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So let’s say it has to do with an axes system. Do these words ring a bell? Yes. Can you sketch here an axes system?

[Nir sketches axes with notches from –10 to 10.] Int: N: Int: N:

So now these axes. . . do you remember them having names, perhaps? I think it was. . . x and y? [Writes x and y near the horizontal and the vertical axes, respectively]. So now if I ask you, in this axes system, to describe the function y=2x, do you have an idea? Yes. So all ah. . . that is if I move here one. . . the y is twice as large as x. So each step in y will give me half of a step in x. Two x’s together will give me one y. So if one y will be 10, then two x’s together will also be 10. So if in this plane [sic] I go up [on the y-axis] and I go up to 10, then one x here will give me 5 and two x’s will give me 10... yea, yea. So the line will go in such a way that one y will give me two x’s. So if it’s one here and two here, this line won’t be 45 degrees, but it will be like this [draws a sketch, see Figure 14] [. . .] If I’ll go 45 degrees, they’ll be equal to one another. The more I tilt the angle, the change will be, in this case of tilting towards this direction [points towards the x-axis] the y in relation to the x will be larger, and if the opposite then the y will be smaller in relation to x. The more I tilt it the more x’s will be equal to one y, according to the angle.

Note that although Nir began with a correct analysis of the ratio between x and y, saying that “each step in y will give half of a step in x”, nonetheless the analysis is later replaced with the proportional misconception pointed out earlier in Category II. Generally, as a response to the interviewer’s mentioning of axes, Nir shifted from an idiosyncratic idea typical of Category VI, to the domain of mathematical conventions. He now treats the function y=2x holistically, estimating its behavior in a manner typical of Category III. The next question posed to Nir was meant to examine this holistic view in a case of a function with an additional free term. Int:

What about if I give you the function y=2x+3, what will you do?

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Figure 14. Nir’s second attempt to draw the function y=2x.

N:

One y we say is equal to two x’s plus 3. Ah. . . how do I treat this 3. . . [pause]. Look, what’s clear to me, without understanding what this means, what this 3 is saying, it’s saying that in principle the y in relation to the x’s is even more. . . [. . .] What this tells me is, that if we add 3 to these x’s, then they will equal one y. This means, that these two x’s are even less, even less than before. So if this was my line before, when y equals 2x, then the line should be even flatter, because to the 2x you have to add 3 for it to be equal to y. Now, how to combine between the number and the x’s, this I can’t figure out. I think I know that the line now is. . . is. . . its angle is smaller in relation to the x-axis by 3. Now, this 3, the question is how it relates to the x’s. What I feel, this is unpleasant, like the tools are gone. I don’t feel like the brain has weakened, but I feel as if I need to do a job but I haven’t got what I need to do it.

At this stage Nir was stuck; according to his analysis he now had to reduce the angle between the line y=2x+3 and the x-axis, compared to the angle of y=2x, but he did not know how to perform this in a way that will reflect the number 3. Here the interviewer intervened again:

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Figure 15. Illustration of Nir’s sketch of the function y=2x+3. The original sketch appears as part of Figure 16.

Int:

N: Int: N:

I see. So, according to your method, if we are to challenge you a bit, all the functions of this sort will go through the origin, that is (0,0). Yes. And this is. . . It’s not right, no, I can’t accept this. So what this is saying is that the reference begins from 3 [draws new axes and sketches the graph of y=2x again]. If we got the y=2x and started it from here, now we actually start it from here [draws a parallel line to y=2x through (3,0), see illustration in Figure 15], and this is our point of reference, yes.

Investigating Nir’s new direction of thought, the interviewer now introduced another task. Two new linear graphs were drawn in the same coordinate system, and Nir was requested to state their equations. The first line was parallel to y=2x and y=2x+3 as drawn by Nir, but ‘starting’ from (0,3). To find its equation, Nir extended the line to its x-intercept, estimating that point as negative six. He then concluded that the equation was y=2x–6. The second line was through (0,4) and (4,0). Nir speculated that the equation of this line was y+4=x+4. Both lines are shown in Figure 16. The discussion about linear graphs ended with Nir’s expression of uncertainty about the last graph: N: This is a bit confusing. I’m not sure, perhaps y+4=x+4? But I think I’m wrong. Int: Why do you think you’re wrong? N: It seems logical, but. . . This is the y plane, and we started from plus 4, and this is the x plane, and we started it from 4 also. . . This is what I can figure out but I might be wrong.

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Figure 16. Nir’s sketches of the functions y=2x and y=2x+3, and his equations for two other given graphs.

The conversation with Nir exemplifies a process of constructing a new schema for solving a problem, when the old schema, acquired many years ago, is no longer accessible. As in Tamar’s case, Nir uses his common sense as a substitution for absent recollections. However, while Tamar continually suggested ideas that would agree with her new schema, in Nir’s case we can notice that ‘memory flashes’, evoked by minor triggers, alter the ongoing schema construction. Thus, in the first stage Nir draws the function using geometrical shapes, knowing that this is not the expected answer. Eliciting the recollection of a coordinate system leads to dismissal of this direction and concentration on the relationship between x and y. In the second stage Nir perceives x and y as quantities whose ratio is expressed by the line’s angle of inclination: As y equals more x’s, the line gets flatter, and thus y=2x+3 is flatter than y=2x. Drawing Nir’s attention to the consequence of this method, i.e., that all lines will pass through the origin, creates a contradiction with another ‘memory flash’, and Nir shifts to the third stage of the construction. In this stage Nir performs translations of y=2x according to the x-intercept, and constructs the idea that the xvalue of the intercept is the number to be added on the right hand side of the equation. Nir extends this line of thought quite coherently, conjecturing about a line with a negative slope. Now the y-intercept is also taken into account, the y-value of this point being added to the left hand side of the equation. Yet, Nir is not satisfied with this idea and expresses his feeling

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of error. This is an example of the influence of a memory, existing even when it cannot be elicited, when one does not recall the right answer but can identify a wrong answer, sensing that ‘this is not it’. 4.3.3. The case of Eli Eli is 40 years old, a principal of a post-secondary school for adults. In high school, he studied mathematics in the low-level track, but quit mathematics classes in twelfth grade, due to what he explained as ‘lack of interest’. After the army service he completed his mathematics matriculation exam and his final grade was 90. Eli’s sketch of the function y=2x was classified within Category II, and was very similar to Shaul’s drawing, shown earlier in Figure 1. Eli obtained his drawing by marking the point (2,1) and tracing a straight line through the origin and this point. In his view, he explained, a line is “a point in motion”. Therefore, in order to draw a line, one has to find the direction of motion, which is set by the function. Eli said: E:

How can you define a line, you want to define its direction. I gave it some direction that was, like, determined by this function. That’s all. . . As far as I’m concerned, the first point has in fact determined the direction.

Eli was asked two more questions. First, he was requested to explain how he obtained the point (2,1). In response, he wrote y=2x and substituted 1 for y, forming the equation 1=2x. Eli solved this equation, not without difficulty, and obtained x=1/2. He then said: E:

So this contradicts my whole theory about this thing. It is something else. So, if I actually go like this. . . now, if I define this as the y-axis, I solved the problem [that is, Eli suggests switching the names of the axes]. [. . .] You see, because then I stay with the same. . . not that it matters. Actually it does matter. Because the minute this is y, then progress of one unit here means progress of half a unit here.

We see here a creative solution to the discrepancy that Eli discovered in regard to his first figure. Since the point x=1/2, y=1 did not fit the line already drawn, and Eli apparently knew that it should, he suggested renaming the x-axis ‘y’ [and vice versa]. This suggestion is, in my opinion, not only flexible but also indicative of a certain capability to look beyond mathematical conventions into the general idea, which in this case is the relationship between two variables. Eli used his common sense, and his solution can be considered no less rational than the expected and conven-

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tional one, i.e., drawing a new line through (1/2,1). We will now see that Eli continued to use his common sense when elaborating on his idea of a line as a point in motion. The second question posed to Eli was to draw the function y=x+3. Eli substituted 0 for y, and then substituted 1 for y. Again, solving the equations 0=x+3 and 1=x+3 turned out to be a complicated task for Eli, who at first did not seem to take into account the possibility of negative x’s. After quite a long process of trial and error, and with the assistance of the interviewer, Eli finally arrived at the points (–3,0) and (–2,1). I will not go into details about this process here. The main point of interest at this stage is, what are Eli’s actions once he marked these points on an axes system. Eli started to connect these two points with a straight line, but changed his mind, and connected the origin with (–2,1), as shown in Figure 17. The following conversation evolved: E: Int: E: Int: E: Int: E: Int:

E: Int: E:

Int: E:

y equals 1 and x equals –2, so it goes here. What about the other point, when y equals 0 and x equals –3? It was here. Yes. . . What happens to the graph? This is the graph [connects the origin with (–3,0), see bold segment on the x-axis in Figure 17]. You’re making three suggestions. Two. First you said it might go like this [refers to the beginning of the line connecting (–3,0) and (–2,1)], then you said it will go like this [the segment between the origin and (–2,1)], and now you’re saying it goes like this [the segment between the origin and (–3,0)]. That’s right. So I don’t really understand if it’s all three or one of the suggestions. At first, with the first given point, it did go like this [the segment on the x-axis]. In my opinion, when the y equals zero we get one given figure. When the y equals 1 then the graph begins to rise, actually. I see. Do you mean that this [the first regretted line] is not. . . No. This cannot be.

After marking down two points obeying the rule of the function, Eli’s first and spontaneous intention was to connect these two points with a straight line. It seems as if a shred of recollection from the learned material has

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Figure 17. Eli’s sketch of the function y=x+3.

emerged as a flashbulb, but Eli immediately rejected it, in light of his newly reconstructed idea of graphing straight lines according to a given direction. It is interesting to note that the result of this idea – that each given point creates a different graph to the same function – does not bother Eli and does not seem to cause any conflict with prior knowledge. On the contrary, it suits his view of a graph as a point in motion, a dynamic and alternating entity. This image is articulated in Eli’s words “the graph begins to rise”. Again, we see here an example of a reconstructed schema that attains dominance over an old learned one.

5. G ENERAL SUMMARY

Investigating the long-term maintenance of knowledge learned in school is a challenging and fascinating pursuit. As Semb and Ellis (1993) noted, “theory development in this area is in its infancy” (p. 310), and therefore there is a potential significance in every effort to broaden the general picture emerging from the existing body of research. In this study I tried to exploit the power of qualitative methods, in order to gain some insight about how adults think when confronted with tasks related to mathematical material learned many years ago. Specifically, the task reported in this paper was to graph simple linear functions. The data, obtained through interviews, was rich and intriguing. As can be expected when it comes

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to human behavior, each case was unique in its own way, and demonstrated a different path of handling the task. Nevertheless, some shared characteristics made it feasible to form categories of recall. The findings reported in section 4 above, support, in general, fundamental assertions made by several memory researchers. Semb and Ellis (1993) claim, based on their review of studies, that the amount of original learning is a prime determinant of what is remembered. Bahrick and Hall (1991), who investigated specifically the maintenance of high school mathematics content, relate to this factor in detail, listing three elements in regard to the original conditions of learning: the number of mathematics courses taken by the student, the total length of the period in which mathematics was learned, and the highest level of the mathematics courses attained. In other words, the preservation of knowledge was found to be dependent on a certain threshold of exposure to the content. In the present study, it was found that graduates of the higher level tracks recalled graphical descriptions of linear functions better than graduates of the low level track. While the former group tended to produce linear graphs, however incorrectly placed, the latter group tended to create idiosyncratic solutions to the task. If we take yet a stricter view of the results, the fact is that most of the subjects were unable to draw a correct sketch of a simple linear function. Of the seven subjects who succeeded in this task, five were graduates of the highest level track. However, the dichotomy of correct / incorrect was not the major theme of this study. Indeed, I believe that the term ‘correct’ is by itself problematic, especially in open settings where actions and responses are susceptible to different interpretations. The documentation of the interactions with the subjects around the issue of graphing functions revealed phenomena of recall that, in my opinion, are beyond the question of ‘to remember or not to remember’ what the function should look like. Such phenomena were demonstrated by presenting three cases, those of Tamar, Nir and Eli. All three have passed the matriculation exam in mathematics with high grades. It is reasonable to assume that in the past they knew how to create a graph of a linear function such as y=2x. Their present attempts to perform this task differed very much from each other. Yet, there was a salient resemblance in those efforts: All three recruited their common sense and tried to present actions that would adhere to some logical framework. They created schemas by which they could explain their actions. Using Bartlet’s (1932) terms, they did not reproduce the learned material, but rather reconstructed it and in fact formed their own personal versions of representing a function in a graph. Acquainted with the basic notions of points and axes, each of them ‘filled the gap’ of the forgotten material with a new, but quite

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coherent, chain of ideas. As a concluding remark, I would like to offer an optimistic look upon these actions. In spite of the evident loss of basic mathematical principles, perhaps the flexibility and freedom to play with ideas in a rational way is the gain that some people receive from their mathematics lessons, rather than the information itself. However, the question remains if this optimistic view could still be held, if the whole range of high school graduates were to be investigated, not just the upper highly educated sector. This is yet to be explored.

ACKNOWLEDGEMENTS The research reported here is part of a doctoral dissertation at the Hebrew University of Jerusalem. I wish to thank Shlomo Vinner for his insightful and supportive instruction.

N OTES 1. The term episodic memory is used here according to Tulving’s original definition, i.e., as referring to recalling personal experiences. In time, however, the use of this term expanded to include other phenomena of recall (see Brewer, 1986, p. 33). 2. These ages can be regarded as forming the ‘settling down’ phase in the adult life (see Levinson, 1978). 3. In Israel mathematics is a compulsory subject throughout high school, and can be studied at three levels, here referred to as high level, intermediate level and low level. A substantial difference exists between the low level and the other two, in terms of curriculum and number of hours alloted to mathematics at school. The difference between the intermediate level and the high level is smaller: In most cases, students at both of these levels study the same curriculum, with more complicated exercises given in the high level classes. The annual report of the Israeli Central Bureau of Statistics usually refers to the intermediate level and the high level together, in distinction from the low level. 4. The candidates’ professions were categorized by The Standard Classification of Occupations, a scale of 10 categories published by the Israeli Central Bureau of Statistics. It is noteworthy that 83% of these 105 professions were classified within the top three categories. 5. This name is a pseudonym, as are all the other subjects’ names.

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R EFERENCES Bahrick, H.P.: 1979, ‘Maintenance of knowledge: Questions about memory we forget to ask’, Journal of Experimental Psychology: General 108(3), 296–308. Bahrick, H.P. and Hall, L.K.: 1991, ‘Lifetime maintenance of high school mathematics content’, Journal of Experimental Psychology: General 120(1), 20–33. Bartlett, F.C.: 1932, Remembering: A Study in Experimental and Social Psychology, Cambridge University Press, Cambridge. Brewer, W.F.: 1986, ‘What is autobiographical memory?’, in D.C. Rubin (ed.), Autobiographical Memory, Cambridge University Press, Cambridge, pp. 25–49. Brewer, W.F. and Nakamura, G.V.: 1984, ‘The nature and functions of schemas’, in R.S. Wyer and T.K. Srull (eds.), Handbook of Social Cognition, volume 1, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 119–160. Cooper, B. and Dunne, M.: 2000, Assessing Children’s Mathematical Knowledge, Open University Press, Buckingham. Kaput, J. and Sims-Knight, J.: 1983, ‘Errors in translations to algebraic equations: Roots and implications’, Focus on Learning Problems in Mathematics 5(3&4), 63–78. Karsenty, R. and Vinner, S.: 1996, ‘To have or not to have mathematical ability, and what is the question’, Proceedings of the 20th International Conference, Psychology of Mathematics Education, Vol. 3, University of Valencia, Valencia, pp. 177–184. Karsenty, R. and Vinner, S.: 2000, ‘What do we remember when it’s over? Adults recollections of their mathematical experience’, Proceedings of the 24t h international Conference, Psychology of Mathematics Education, Vol. 3, Hiroshima University, Hiroshima, pp. 119–126. Levinson, D.J.: 1978, The Seasons of a Man’s Life, Alfred A. Knopf, New York. Neisser, U.: 1978, ‘Memory: What are the important questions?’ in M.M. Gruneberg, P.E. Morris and R.N. Sykes (eds.), Practical Aspects of Memory, Academic Press, London, pp. 3–24. Neisser, U.: 1967, Cognitive Psychology, Appleton-Century-Crofts, New York. Neisser, U.: 1984, ‘Interpreting Harry Bahrick’s discovery: What confers immunity against forgetting?’, Journal of Experimental Psychology: General 113, 32–35. Philipp, R.A.: 1992, ‘A study of algebraic variables: Beyond the student-professor problem’, Journal of Mathematical Behavior 11(2), 161–176. Rosnick, P.: 1981, ‘Some misconceptions concerning the concept of variable’, Mathematics Teacher 74(6), 418–420. Rosnick, P. and Clement, J.: 1980, ‘Learning without understanding: The effect of tutoring strategies on algebra misconceptions’, Journal of Mathematical Behavior 3(1), 3–27. Semb G.B. and Ellis, J.A.: 1992, ‘Knowledge Learned in College: What is Remembered?’ Paper presented at the annual meeting of the American Educational Research Association, San Francisco. Semb, G.B., Ellis, J.A. and Araujo, J.: 1993, ‘Long-term memory for knowledge learned in school’, Journal of Educational Psychology 85(2), 305–316. Skemp, R.: 1976, ‘Relational understanding and instrumental understanding’, Mathematics Teaching 77, 20–26. Stake, R.E.: 1994, ‘Case studies’, in N.K. Denzin and Y.S. Lincoln (eds.), Handbook of Qualitative Research, Sage, Thousand Oaks, CA, pp. 236–247. Stake, R.E.: 1995, The Art of Case Study Research, Sage, Thousand Oaks, CA. Tulving, E.: 1972, ‘Episodic and semantic memory’, in E. Tulving and W. Donaldson (eds.), Organization of Memory, Academic Press, New York, pp. 381–403.

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Vinner, S.: 1983, ‘Concept definition, concept image and the ion of function’, International Journal of Mathematics Education in Science and Technology 14(3), 293–305.

Weizmann Institute of Science, Department of Science Education, Rehovot 76100 Israel, Telephone 972-89343073, Fax 972-89344174

BOOK REVIEW

Barbin, E., Duval, R., Giorgiutti, I., Houdebine, J. and Laborde, C. (Eds.), Produire et lire des textes de démonstration, Paris: Ellipses, 2001, ISBN: 2-7298-0675-X, 266 pp. This edited collection focuses on the processes of producing and interpreting the ‘texts’ – in a broad sense – through which the unfolding of mathematical proofs is made explicit. The guiding idea of the book is that such texts can be viewed from many perspectives; not only mathematical, but historical, epistemological, didactic and cognitive. The essentials are well captured in the opening collective chapter by Barbin, Duval, Houdebine and Laborde which analyses – from each of these perspectives – three proof texts relating to a particular geometric property. The property in question is the trisection of the diagonal between two vertices of a parallelogram by a line segment drawn from one of the other vertices to the midpoint of an opposite side. One of the considerations guiding the authors’ choice of this property as an example is its amenability to treatment in terms of different geometrical frameworks: classical, vectorial and barycentrical1 . The particular proof texts chosen for analysis come from a contemporary schoolbook and a well-known early-twentieth-century collection of exercises and problems. The opening epistemological analysis “with a historical base” considers the texts in terms of their dependence on method of reasoning: one of the texts relies on vectorial and algebraic calculation; the others on a coordinated reading of written text and accompanying figure; in particular, on identifying elements referred to in the written text within a complex figure. A simpler proof is suggested – based on operations with triangle pieces and calculations concerning these – on the principle of reducing or eliminating the “logico-discursive”. The following cognitive analysis 1 ‘Barycentrical’ is a term not familiar in English. As used here, it refers to geomet-

rical arguments that rest on the taken-as-given properties of the centroid of a triangle; and generalises to include similar use of the properties of other geometrical – and physical – centres. (For the english term ‘barycentre’, ‘the centre of mass of a system’, see . For the French term ‘barycentre’, see . Educational Studies in Mathematics 51: 145–147, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

146 examines the place of individual propositions of the written text within the overall flow of argument, and the shifts of attention between written text and accompanying figure. The main conclusion is that the complexity of the texts depends not on the passage from one propositional step to the next but on the limitations of designating in words those objects within the figure that a proposition refers to. The third analysis focuses on the differential framing of the task to which the texts constitute responses; as problem-to-solve or as result-to-prove; as a matter of computation or of reasoning. The closing didactical analysis shifts the focus towards matters of teaching choices. The three texts are seen as having the potential to give insight into the multiplicity of mathematical styles. But perhaps most important, much remains implicit in the texts as presented; in particular, the heuristic reasoning informing the development of arguments is not signalled. From these four contributions, the “paradoxical” conclusion is drawn: “If a proof is never identifiable in the text which presents it, it cannot be made without a text which makes explicit and accomplishes the different operations of the course of the proof” (p. 8). The remainder of the book is organised into five sections. The first of these is entitled “epistemological and historical aspects”. Here, Barbin compares approaches to writing proofs, with the aim of capturing the way in which mathematicians use words and images. Guichard examines ten proof texts of the same geometrical theorem authored by what he terms ‘pedagogue mathematicians’, and then five further proof texts chosen for their very different styles. The second section is entitled “proof as an object of teaching”. Beck reports a linguistic analysis comparing proofs in school mathematics textbooks with other argumentative texts (ranging from Conan Doyle to Voltaire) encountered by pupils at a similar age. Houdebine compares a broader range of mathematical proof texts – from algebra, analysis, and probability as well as geometry – showing that despite their apparent diversity the underlying rules of organisation are essentially the same, with differences relating more to variations in the frequency of particular elements. The third section is entitled “teaching sequences”. Bellard and Lewillion discuss the use of visual schemas intended to help pupils distinguish premises from conclusion in theorem statements, and then analyse related examples of pupils’ work. Combes and Bonafé analyse pupils’ written “research narratives” recording their strategic moves in investigating a simple combinatorial problem by generating a data pattern with a view to forming an algebraic generalisation. Thomas-Van Dieren reports a teaching approach in which practical drawing and colouring activities provide a basis

147 for subsequent mathematical analysis of these activities and the structure of the materials they involve. The fourth section is entitled “pupils and proof”. Duval presents an extended theorisation of the distinctive cognitive features of writing which he summarises as follows: two types of rationality have developed in western thought; one is linked to speech, based on dialogue, and oriented towards the regulation of social interactions; the other, developed much later, is linked to writing, meeting the needs of control and proof, and oriented towards the creation of theoretical models. On this basis he argues that learning mathematical proof makes a contribution to the general education of pupils by giving them access to these types of rationality, and developing their awareness of the differences between them. Houdebine describes – and exemplifies the application of – a framework for analysing pupils’ productions with the aim of identifying those elements relating to proof, and then diagnosing the strengths and weaknesses of individual pupils with a view to helping progress. The fifth and final section is entitled “the computer tool”. Its focus is on the treatment of proof texts in intelligent tutoring systems for geometry. Py provides a general introduction, followed by specific discussion of Mentoniezh and ARRIA. Luengo discusses the design rationale of Cabri-Euclide, linked to some of the ideas considered in earlier sections. Rather briefly, Simon describes Premiers Pas, and Py gives a summary of some work with Mentoniezh. El Gass and Giorgiutti report on pupils’ interactions with DEFI, and compare pupils’ productions in computer and pencil-and-paper environments. Disappointingly, there is no concluding chapter to synthesise the varied individual contributions to the book, to suggest their collective implications for policy and practice, and to relate them back to the wider research field. As far as these latter matters are concerned, few of the contributions are strongly oriented towards policy or practice. Equally, little account is taken of wider bodies of work in mathematics education on the teaching and learning of mathematical proof, or on issues of language and representation, reading and writing. However, for those seeking new contributions to the evolving scholarship of proof in mathematics education, this book offers some interesting lines of theorisation and distinctive approaches to analysis. K ENNETH RUTHVEN University of Cambridge, Faculty of Education, Cambridge, United Kingdom

WERNER BLUM ET AL.

ICMI STUDY 14: APPLICATIONS AND MODELLING IN MATHEMATICS EDUCATION – DISCUSSION DOCUMENT

ABSTRACT. This paper is the Discussion Document for a forthcoming ICMI Study on Applications and Modelling in Mathematics Education. As will be well-known, from time to time ICMI (the International Commission on Mathematical Instruction) mounts specific studies in order to investigate, both in depth and in detail, particular fields of interest in mathematics education. The purpose of this Discussion Document is to raise some important issues related to the theory and practice of teaching and learning mathematical modelling and applications, and in particular to stimulate reactions and contributions to these issues and to the topic of applications and modelling as a whole (see Section 4). Based on these reactions and contributions, a limited number (approximately 75) of participants will be invited to a conference (the Study Conference) which is to take place in February 2004 in Dortmund (Germany). Finally, using the contributions to this conference, a book will be produced (the Study Volume) whose content will reflect the state-of-the-art in the topic of applications and modelling in mathematics education and suggest directions for future developments in research and practice. The authors of this Discussion Document are the members of the International Programme Committee for this ICMI Study. The committee consists of 14 people from 12 countries, listed at the end of Section 4. The structure of the Document is as follows. In Section 1, we identify some reasons why it seems appropriate to hold a study on applications and modelling. Section 2 sets a conceptual framework for the theme of this Study, and Section 3 contains a selection of important issues, challenges and questions related to this theme. In Section 4 we describe possible modes and ways of reacting to the Discussion Document, and in the final Section 5 we provide a short bibliography relevant to the theme of this Study.

1. R ATIONALE FOR THE STUDY

Among the themes that have been central to mathematics education during the last 30 years are relations between mathematics and the real world (or better, according to Pollak, 1979, the ‘rest of the world’). In Section 2.1 we shall deal with terminology in more detail but, for the moment, we use the term ‘applications and modelling’ to denote any relations whatsoever between the real world and mathematics. Applications and modelling have been an important theme in mathematics education as can be seen from the wealth of literature on the topic, and from material generated from a multitude of national and international conferences. Let us mention, in particular, firstly the ICMEs Educational Studies in Mathematics 51: 149–171, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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(the International Congresses on Mathematical Education) with their regular working or topic groups and lectures on applications and modelling, and secondly the series of ICTMAs (the International Conferences on the Teaching of Mathematical Modelling and Applications) which have been held biannually since 1983. Their Proceedings and Survey Lectures (see the bibliography in Section 5) indicate the state-of-the-art at the relevant time and contain many examples, studies, conceptual contributions and resources addressing the relation between the real world and mathematics, for all levels and all institutions of the educational system. In curricula and textbooks we find many more relations to real world phenomena and problems than, say, ten or twenty years ago. While applications and modelling also play a more important role in most countries’ classrooms than in the past, there still exists a substantial gap between the ideals of educational debate and innovative curricula, on the one hand, and everyday teaching practice on the other hand. In particular, genuine modelling activities are still rather rare in mathematics lessons. Altogether, during the last few decades there has been a lot of work in mathematics education centered on applications and modelling. The primary focus of many activities was on practice, e.g. on constructing and trying out mathematical modelling examples for teaching and examinations, writing application-oriented textbooks, implementing applications and modelling into existing curricula or developing innovative, modellingoriented curricula. Several of these activities contained research components as well if (according to Niss, 2001) we consider research as “the posing of genuine, non-rhetorical questions . . . to which no satisfactory answers are known as yet . . . and . . . the undertaking of non-trivial investigations of a systematic, reflective and ‘methodologically conscious’ nature” in order to obtain answers to those questions. In this sense, there are specific applications and modelling research activities, such as: clarification of relevant concepts; investigation of competencies and identification of difficulties and strategies activated by students when dealing with application problems; observation and analysis of teaching, and study of learning and communication processes in modelling-oriented lessons; evaluation of alternative approaches used to assess performance in applications and modelling. In particular, during the last few years the number of genuine research contributions has increased as can be seen in recent ICTMA Proceedings. It is not at all surprising that applications and modelling have been – and still are – a central theme in mathematics education. Nearly all questions and problems in mathematics education, that is questions and problems concerning human learning and teaching of mathematics, affect

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and are affected by relations between mathematics and the real world. For instance, one essential answer (of course not the only one) to the question as to why all human beings ought to learn mathematics is that it provides a means for understanding the world around us, for coping with everyday problems, or for preparing for future professions. When dealing with the question of how individuals acquire mathematical knowledge, we cannot get past the role of relations to reality, especially the relevance of situated learning (including the problem of the dependence on specific contexts). The general question as to what, after all, ‘mathematics’ is, as part of our culture and as a social phenomenon, of how mathematics has emerged and developed, points also to ‘applications’ of mathematics in other disciplines, in nature and society. Today mathematical models and modelling have invaded a great variety of disciplines, leaving only a few fields where mathematical models do not play some role. This has been substantially supported and accelerated by the availability of powerful electronic tools, such as calculators and computers with their enormous communication capabilities. In the current OECD (Organisation for Economic Co-operation and Development) Study PISA (Programme for International Student Assessment), relations between the real world and mathematics are particularly topical. What is being tested in PISA is ‘mathematical literacy’, that is (see the PISA mathematics framework in OECD, 1999) “an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgements and to engage in mathematics, in ways that meet the needs of that individual’s life as a constructive, concerned, and reflective citizen.” That means the emphasis in PISA is “on mathematical knowledge put into functional use in a multitude of different situations and contexts”. Therefore, mathematising real situations as well as interpreting, reflecting and validating mathematical results in ‘reality’ are essential processes when solving literacy-oriented problems. Following the 2001 publication of results of the first PISA cycle (from 2000), an intense discussion has started, in several countries, about the aims and the design of mathematics instruction for schools, and especially about the role of mathematical modelling, applications of mathematics and relations to the real world. In mounting this Study on ‘Applications and Modelling in Mathematics Education’, ICMI takes into account the above-mentioned reasons for the importance of relations between mathematics and the real world as well as the contemporary state of the educational debate, of research and development in this field. This does not, of course, mean that we already know all answers to the essential questions in this area and that it is merely

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a matter of putting together these answers in the Study. Rather, it is an important aim of the Study to identify the shortcomings and to stimulate further research and development activities. Nevertheless, it is time to map out the state-of-the-art in theory and practice, in research and development of applications and modelling in mathematics education, and to document these in this Study. Documenting the state-of-the-art in a field and identifying deficiencies and needed research requires a structuring framework. This is particularly important in an area which is as complex and difficult to survey as the teaching and learning of mathematical modelling and applications. As we have seen, this topic not only deals with most of the essential aspects of the teaching and learning of mathematics at large, but it also touches upon a wide variety of versions of the real world outside mathematics that one seeks to model. Perceived in that way, the topic of applications and modelling may appear to encompass all of mathematics education plus a lot more. It is evident, therefore, that we have to find a way to conceptualise the topic so as to reduce the complexity to a meaningful and tractable level. In the following Section 2 we offer our conceptualisation of the topic: in section 2.1 we clarify some of the basic concepts and notions of the field, and in section 2.2 we suggest a structure for the field. This serves as a basis for identifying important challenges and questions in Section 3, the core of this Discussion Document.

2. F RAMEWORK FOR THE STUDY

2.1. Concepts and notions In this section we give, in a rather pragmatic way, some working definitions that will be useful for the following sections. This is not the place for a deeper epistemological analysis of these concepts. Rather, this can be done in the Study itself. By real world we mean everything that has to do with nature, society or culture, including everyday life as well as school and university subjects or scientific and scholarly disciplines different from mathematics. For a description of the complex interplay between the real world and mathematics we use one of the well-known simple models developed for that purpose (see Blum/Niss, 1991, and the literature quoted there). The starting point is normally a certain situation in the real world. Simplifying it, structuring it and making it more precise – according to the problem solver’s knowledge and interests – leads to the formulation of a problem and to a real model of the situation. The term ‘problem’ is used in a broad sense

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here, encompassing not only practical problems but also problems of a more intellectual nature aiming at describing, explaining, understanding or even designing parts of the world. If appropriate, real data are collected in order to provide more information about the situation at one’s disposal. If possible and adequate, this real model – still a part of the real world in our sense – is mathematised, that is the objects, data, relations and conditions involved in it are translated into mathematics, resulting in a mathematical model of the original situation. Now mathematical methods come into play, and are used to derive mathematical results. These have to be re-translated into the real world, that is interpreted in relation to the original situation. At the same time the problem solver validates the model by checking whether the problem solution obtained by interpreting the mathematical results is appropriate and reasonable for his or her purposes. If need be (and more often than not this is the case in ‘really real’ problem solving processes), the whole process has to be repeated with a modified or a totally different model. At the end, the obtained solution of the original real world problem is stated and communicated. The process leading from a problem situation to a mathematical model is called mathematical modelling. However, it has become common to use that notion also for the entire process consisting of structuring, mathematising, working mathematically and interpreting/validating (perhaps several times round the loop) as just described. Sometimes the given problem situation is already pre-structured or is nothing more than a ‘dressing up’ of a purely mathematical problem in the words of a segment of the real world. This is often the case with classical school word problems. In this case mathematising means merely ‘undressing’ the problem, and the modelling process only consists of this undressing, the use of mathematics and a simple interpretation. Using mathematics to solve real world problems is often called applying mathematics, and a real world situation which can be tackled by means of mathematics is called an application of mathematics. Sometimes the notion of ‘applying’ is used for any kind of linking of the real world and mathematics. During the last decade the term ‘applications and modelling’ has been increasingly used to denote all kinds of relationships whatsoever between the real world and mathematics. The term ‘modelling’, on the one hand, focuses on the direction from reality to mathematics and, on the other hand and more generally, emphasizes the processes involved. The term ‘application’, on the one hand, focuses on the opposite direction from mathematics to reality and, on the other hand and more generally, emphasizes the objects involved – in particular those parts of the real world

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which are accessible to a mathematical treatment and for which there exists corresponding mathematical models. It is in this comprehensive sense that we understand the term ‘applications and modelling’ as used in the title of this Study. 2.2. Structure of the topic Applications and Modelling in mathematics education Let us begin by addressing what one may refer to as ‘the reality’ of applications and modelling in mathematical education. We think of this reality as being constituted essentially by two dimensions: The significant ‘domains’ within which mathematical applications and modelling are manifested, on the one hand, and the educational levels within which applications and modelling may be taught and learnt, on the other hand. More specifically, in the first dimension we discern three different domains, each forming some sort of a continuum. The first domain consists of the very notions of applications and modelling, i.e. what we mean by an application of mathematics, and by mathematical modelling; what are the most important components of applications and modelling, in terms of concepts and processes; what are the epistemological characteristics of applications and modelling, in relation to mathematics as a discipline and other disciplines and areas of practice; who uses mathematics, and for what purposes, and with what sorts of outcomes; what is modelling competency, etc. The second domain is that of the classroom. We use this term as a broad indicator of the location of teaching and learning activities pertaining to applications and modelling. Of course, this includes the classroom in a literal sense, but it also includes the student doing his or her homework, individually or in groups, and the teacher’s planning of teaching activities or looking at students’ products, written or other, and so forth. The third and final domain is the system domain. The word system, here, refers to the whole institutional, political, structural, organisational, administrative, financial, social, and physical environment that exerts an influence on the teaching and learning of applications and modelling. It appears that we have chosen not to consider individuals, in particular students and teachers, as constituting separate domains. This does not imply, however, that individuals are not part of our conceptualisation. The individual student is a member of the classroom, as defined above, when engaging in learning activities in applications and modelling. The individual teacher can also be regarded as a member of the applications and modelling classroom, namely when he or she is engaged in teaching, supervising, advising or assessing students. From another perspective, however, the teacher is also a member of the system. This happens when he or she speaks or acts on behalf of

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the system (typically in the form of his or her institution) in matters concerning selection, placement, and examination of the individual student, or invokes rules, procedures or other boundary conditions in decisions on, say, curricular matters. The second dimension is constituted by the educational levels at which applications and modelling are taught and learnt. We have decided to adopt a relatively crude division of levels, both in order to avoid excessive detail in the discussion document and in order to obtain a division which is compatible with the state of affairs in most, maybe all, countries in the world. The levels adopted are the primary, the secondary, the tertiary levels, and the level of teacher education. We do not primarily refer to age levels here, but rather to the intrinsic levels of the learners’ knowledge and competencies. It goes without saying that a much more fine-grained division might have been an alternative. Nevertheless, at least the present one does allow for the consideration of applications and modelling at all educational levels, albeit possibly within larger clusters of rather diverse kinds of education. We might, of course, design a more subtle structure and regard, for instance, the system as an independent dimension. However, for our purposes it is quite appropriate to represent the ‘reality’ of applications and modelling in mathematics education by the Cartesian product of those two dimensions, the domains and the levels. If we do so then a major point in this framework is to identify and consider a number of crucial issues concerning that ‘reality space’. An issue addresses a segment of reality, which can be represented as a certain subset of the reality space. That subset consists of those objects, phenomena or situations – drawn from combinations of applications and modelling domains and educational levels – that the issue concerns. To illustrate the point, let us give but one example of such an issue: Issue 0: One of the underlying reasons for attributing a prominent position to applications and modelling in the teaching and learning of mathematics is that it is assumed desirable for students to be able to engage, outside of the mathematics classroom, in applications and modelling concerning areas and contexts that are new to them. In other words, it is assumed that applications and modelling competency developed in and for some types of areas and contexts can be transferred to other such types having different properties and characteristics. However, many research studies suggest that for some categories of students this transfer is rather limited in scope and range. To what extent is applications and modelling competency transferable across areas and contexts? What teaching/learning experiences are needed or suitable to foster such transferability?

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As it stands here, this issue concerns the classroom domain and (at least) the primary, secondary, and tertiary levels. This means that the issue addresses the ‘rectangle’ constituted by the entire classroom domain and the first three educational levels. If for some reason the focus were to be limited to address, say, the secondary level, then of course the rectangle would be reduced accordingly. In this Discussion Document, a number of issues have been identified as particularly significant to the present study. Readers are invited to comment on these issues or to suggest further issues to be considered in the Study. Each of the issues addresses its own segment of reality, but of course those segments may intersect. The reality space and the issues addressing it may be mapped as in Figure 1.

Figure 1. The ‘reality’ of applications and modelling.

The reader may have noticed that the formulation of the issue given above as an example consists of two parts. Firstly, a background part outlining a challenge, i.e. a dilemma or a problem which may be of a political, practical, or intellectual nature. For short let us call this part the challenge part of the issue. The second part consists of particular questions that serve the purpose of pinpointing some crucial aspects of the challenge that deserve to be dealt with in the Study. When the Study has been completed, a

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substantial portion of it will consist of analyses of a number of significant issues. From the point of view of this Study, an issue concerning applications and modelling in mathematics education may be viewed and approached – depending on its nature – from a variety of different perspectives, each indicating the category of answers sought. The most basic of these perspectives is that of doing, i.e. actual teaching and learning practice as enacted and carried out in the classroom (in the sense defined above). Here, the focus is on what does (or should) take place in existing everyday classrooms at given educational levels. Another perspective is the development and design of curricula, teaching and learning materials or activities, and so forth. Here, the focus is on establishing short or long term plans and conditions for future teaching and learning. A third perspective is that of research, which focuses on the generation of answers to research questions as yet unanswered, while a fourth and final perspective is that of policy for which the focus is on the instruments, strategies and policies that are or ought to be adopted in order to place matters pertaining to applications and modelling on the agenda of practice or research in some desired way. Accordingly, a given issue may be addressed from one or some (perhaps all) of these four perspectives. To avoid a possible misinterpretation let us stress that the order in which we have presented these four perspectives does not imply a hierarchy. It appears that each of these perspectives can be perceived as representing a particular professional role: The role of teacher or student, the role of curriculum developer, the role of researcher, and the role of lobbyist or decision maker. An individual can assume all of these roles, but usually not at the same time. In the above-mentioned example, the issue may be approached from the perspective of ‘doing’, provided the interest and emphasis is on the actual construction of learning environments and the carrying through of specific teaching activities meant to underpin transferability of application and modelling competencies cultivated within certain areas and contexts to other such areas or contexts. The ‘development and design’ perspective is adopted if the emphasis is on finding or devising ways to orchestrate teaching and learning activities that are hoped to generate improved transferability. If, on the other hand, the emphasis is on getting to know and understand the nature and extent of transferability of such competencies between areas and contexts, or the effect of an implemented design, then the ‘research’ perspective is being invoked. Finally, the ‘policy’ perspective is on the agenda if the focus is on pleading or lobbying for, say, making room and time in the curriculum for activities that are seen as necessary or desirable for allowing students to gain multi-faceted and rich first-hand

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experiences with a large variety of applications and modelling activities drawn from different areas, contexts, and situations. We may depict, metaphorically, the different perspectives by a quadrifocal magnifying glass, in which each of the four segments represents a characteristic focus and a corresponding focal length, both defined by the perspective; see Figure 2:

Figure 2. Perspectives on ‘reality’.

So far, we have described the nature of the reality and the issues which are going to be considered in this ICMI Study, and the perspectives through which they can be addressed. One final component in our attempt to structure the Study remains to be identified: the recording of the deliberations and investigations that are going to be conducted in the Study. This Study will be conducted by employing the four perspectives to look at the reality and the issues, and the outcomes of analysing the reality and the issues. Reflecting upon these from various perspectives will provide the substance for what will be recorded in the resulting ICMI Study Volume. Accordingly, that Volume will consist of sections in which the ‘reality’ is described in detail, together with the issues eventually identified, sections in which the different segments of the magnifying glass are polished, and sections in which conclusions concerning the issues raised and dealt with by means of the four perspectives are presented and discussed.

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3. E XAMPLES OF IMPORTANT ISSUES

In this chapter a number of selected issues – consisting of challenges and questions – are raised. Although they have been grouped by certain inherent features, there is much overlap between them and different groupings are certainly possible. They are intended as a guide to the kinds of issues that the present Study intends to address. Readers are invited to come up with additional relevant issues. 3.1. Epistemology Issue 1: There are a number of different elements and characterisations of modelling and applications; some of these are posing and solving open-ended questions, creating, refining and validating models, mathematising situations, designing and conducting simulations, solving word problems and engaging in applied problem solving. All of these link the field of mathematics and the world. If the goal of knowledge is to assist us to ensure the sustainability of health, education and environmental well-being and improve its quality, then individuals must engage in applications and modelling to do so. Which is the description/representation of the components within applications and modelling is relevant to this issue? How is the relationship between applications and modelling and mathematics including its domains, concepts, representations, skills, methods and forms of evidence best described? What is the relationship between applications and modelling and the world we live in?

Examples of specific questions that could be addressed here are: – What are the process components of modelling? What is meant by or involved in each? – How does our knowledge of applications and modelling accumulate, evolve and change over time? – What parts of mathematics, if any, are less likely to be represented in applications and modelling? – What parts of applications and modelling, if any, are less likely to be represented in mathematics? – What is the meaning and role of abstraction, formalisation and generalisation in applications and modelling? – What is the meaning and role of proof and proving in applications and modelling? Are there common features of proving and modelling? – What are the various meanings of ‘authenticity’ in modelling? – How much extra-mathematical context must be familiar and understood to undertake applications and modelling? – What is generalisability and transfer when working across contexts?

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3.2. Application problems There exists a wealth of applications and modelling problems and materials for use in mathematics classrooms at various educational levels. These materials range from mere ‘dressed up’ mathematical problems to ‘really real’, authentic problem situations. Issue 2: An important aspect in the research and practices of applications and modelling is the notion and role of authenticity of the problems and situations dealt with in applications and modelling activities. What does research have to tell us about the significance of authenticity to students’ acquisition and development of modelling competency?

Examples of specific questions: – What authentic applications and modelling materials are available worldwide? – Taking account of teaching objectives and students’ personal situations (experience, competence), how can teachers set up authentic applications and modelling tasks? – How does the authenticity of problems and materials affect students’ ability to transfer acquired knowledge and competencies to other contexts and situations? 3.3. Modelling abilities and competencies With the teaching and learning of mathematical modelling and applications, a great variety of goals and expectations are combined. Issue 3a: One of the most important goals is for students to acquire modelling ability and competency. How can modelling ability and modelling competency be characterised, and how can it be developed over time?

Examples of specific questions: – Are modelling ability and modelling competency different concepts? – Can specific subskills and subcompetencies of ‘modelling competency’ be identified? – How can modelling ability be distinguished from general problem solving abilities? – Are there identifiable stages in the development of modelling ability?

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– What are the characteristic differences between expert modellers and novice modellers? What are characteristic features of the activity of students who have little experience of modelling? – What is the role of pure mathematics in developing modelling ability? – What are common features, and what are differences between students’ individual ability and interactive ability in applications and modelling? An especially important problem is the context dependence of acquired competencies. This holds for modelling competency as well. See Issue 0 in section 2.2! The development of applications and modelling abilities and competencies is of particular relevance for teachers. Issue 3b: It appears rare that mathematics teacher education programmes include orientations to modelling, and the use of the modelling process in mathematics courses. How can modelling in teacher pre-service and in-service education courses be promoted?

Examples of specific questions: – What is essential in a teacher education programme to ensure that prospective teachers will acquire modelling competencies and be able to teach applications and modelling in their professional future? – Considering both the limited mathematics background of primary school student teachers and the limited time available for mathematics in their education, how can they experience real, non-trivial modelling situations? – Which training strategies can help teachers develop security with respect to using applications and modelling in their teaching? 3.4. Beliefs, attitudes, and emotions Beliefs, attitudes and emotions play important roles in the development of critical and creative senses in mathematics. Issue 4: Modelling aims, among other things, at providing students with a better comprehension of mathematical concepts, teaching them to formulate and to solve specific situation-problems, awaking their critical and creative senses, and shaping their attitude towards mathematics and their picture of it. What is the potential of applications and modelling to provide an environment to support both students and teachers in their development of appropriate beliefs about and attitudes towards mathematics?

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Examples of specific questions: – Taking into account available research on the role of beliefs, attitudes and emotions in learning applications and modelling, what are the implications of this research for changing teaching practice and classroom cultures with respect to applications and modelling? – Can modelling effectively contribute towards promoting views of mathematics that extend beyond transmissive techniques to its role as a tool for structuring other areas of knowledge? – What strategies are feasible for in-service teacher education that will address the fear experienced by some teachers when faced with applications and modelling? 3.5. Curriculum and goals Applications and modelling can make fundamental contributions to the development of students’ competencies. This is why it ought to be present in all mathematics curricula. Issue 5a: Serious applications and modelling activities are always time consuming because attention has to be paid to several crucial phases of the modelling process including work on extra-mathematical matters. This implies that applications and modelling components of general mathematical curricula (at whichever level) will have to ‘compete’ with other components of the mathematics curriculum, in particular work on pure mathematics. What would be an appropriate balance – in terms of attention, time and effort – between applications and modelling activities and other mathematical activities in mathematics classrooms at different educational levels?

Examples of specific questions: – What is the actual role of applications and modelling in curricula in different countries? – Is it possible – or desirable – to identify a core curriculum in applications and modelling within the general mathematical curriculum? – Which applications, models and modelling processes should be included in the curriculum? Does the answer depend on each teacher or should there be some minimal indications in national and state curricula? – Is it beneficial to generate specific courses or programmes on applications and modelling or is it better to mix them in the standard mathematical courses? – Is it possible to treat applications and modelling in the curricula as an interdisciplinary activity?

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– When applications and modelling are included at different places in mathematics curricula, how can it be guaranteed that basic modelling skills and competencies are acquired systematically and coherently? The university level represents a particularly problematic case, whence the next issue. Issue 5b: Although there are major differences between different places and countries, university graduates in mathematics (even the most specialised and advanced ones) embark on a large variety of different professional careers, many of which will have links to matters pertaining to applications and modelling. It can be argued that even future research mathematicians are likely to come in touch with applications and modelling in some way or another, perhaps because their research activities will be informed by application problems or because they will be teaching students with application careers in front of them. Should a plea be made for all university graduates in mathematics to acquire some applications and modelling experiences as part of their studies? If so, what kinds of experiences should they be?

Concerning general education at the school level, some special questions arise. Issue 5c: Mathematics accounts for a large proportion of time in school. This is only justified if mathematics can contribute to general education for life after school. How and to what extent can applications and modelling contribute to building up fundamental competencies and to enriching a student’s general education?

Examples of specific questions: – What meanings can be given to ‘general education’, and what is the role of mathematical modelling therein? Are applications and modelling really an indispensable part of ‘general education’ for students? – What pictures do teachers have in their mind about the contribution of mathematics – and in particular of mathematical modelling – to general education, and how can these be influenced? – What is a suitable balance within general education of the following emphases: to create one’s own models of real situations and problems, or to make judgements about models made by others? 3.6. Modelling pedagogy The pedagogy of applications and modelling intersects the pedagogy of pure mathematics in a multitude of ways and requires at the same time a

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variety of practices that are not part of the traditional mathematics classroom. Issue 6: While examples of successful applications and modelling initiatives have been documented in a variety of countries, and contexts, the extent of such programmes remains less than desired. Furthermore approaches to teaching applications and modelling vary from the use of traditional methods and course structures, to those that include a variety of innovative teaching practices including an emphasis on group activity. What are appropriate pedagogical principles and strategies for the development of applications and modelling courses and their teaching? Are there different principles and strategies for different educational levels?

Examples of specific questions: – What research evidence is available to inform and support the pedagogical design and implementation of teaching strategies for courses with an applications and modelling focus? – What are the areas of greatest need in supporting the design and implementation of courses with an applications and modelling focus? – To what extent do teaching practices within applications and modelling courses draw on general theories of human development and/or learning? – What criteria are most helpful in selecting methods and approaches suggested by such theories? – What obstacles appear to inhibit changes in classroom culture e.g. the introduction of interactive group work in applications and modelling? – What criteria can be used to choose (e.g. between individual and group activity) the most desirable option at a particular point within an applications and modelling teaching segment? – What documentations of successful group learning practices exist?

3.7. Sustained implementation To change an educational system is a major challenge as it involves and impacts upon many different parties. Implementing new mathematical modelling curricula involves specific factors, such as in-service teacher training, technological requirements, etc. Sustaining this implementation requires changes in pre-service education, and more general agreement between mathematics faculty members at the post secondary level.

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Issue 7: With the increasing interest in and argument for mathematical modelling both inside and outside the mathematical community, there is a need to ensure that mathematical modelling is implemented in a sustained fashion at all levels of mathematics education. In spite of a variety of existing materials, textbooks, etc., and of many arguments for the inclusion of modelling in mathematics education, why is it that the actual role of applications and mathematical modelling in everyday teaching practice is still rather marginal, for all levels of education? How can this trend be reversed to ensure that applications and mathematical modelling is integrated and preserved at all levels of mathematics education?

Examples of specific questions: – What are the major impediments and obstacles that have existed to prevent the introduction of applications and mathematical modelling, and how can these be changed? – What documented evidence of success in overcoming impediments to the introduction of applications and modelling courses exist? – What are the requirements for developing a mathematical modelling environment in traditional courses at school or university? – How does one ensure that the mathematical modelling philosophy in curriculum documents is mirrored in classroom practice? – What continuing education experiences (and education support for teachers, teaching assistants, mathematics faculty, etc.) need to be provided? 3.8. Assessment and evaluation The teaching and learning of mathematics at all levels is naturally closely related to assessment of student achievement. Nevertheless, to assess mathematical modelling is not easy to accomplish. The more complicated and open a problem is, the more complicated it is to assess the solution, and if one adds the component of available technology, assessment becomes even more complicated. Similar problems are inherent at the course level for the evaluation of programmes with application and modelling components. Issue 8a: There seem to be many indications that the assessment modes traditionally used in mathematics education are not fully appropriate to assess students’ modelling competency. What alternative assessment modes are available to teachers, institutions and educational systems that can capture the essential components of modelling competency, and what are the obstacles to their implementation?

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Examples of specific questions: – What are the possibilities or obstacles when assessing mathematical modelling as a process (instead of a product)? What can be learnt from assessment in the arts, music, etc.? – If there is a change in the mathematics conception of students after experiencing and learning mathematical modelling, how do we assess that change? – In teacher education, what techniques can be used to assess a future teacher’s ability to teach and assess mathematical modelling? – When mathematical modelling is introduced into traditional courses at school or university, how should assessment procedures be adapted? – When centralised testing of students is implemented, how do we ensure that mathematical modelling is assessed validly? – How does one reliably assess individual contributions and achievement within group activities and projects? Issue 8b: There seems to be a need to develop specific means of evaluating programmes with an applications and modelling content. What evaluation modes are available that can capture the essential features of applications and modelling, especially of integrated courses, programmes and curricula, and what are the obstacles to their implementation?

Examples of specific questions: – In what way do usual evaluation procedures for mathematical programmes carry over to programmes that combine mathematics with applications and modelling? – What counts as success when evaluating outcomes from a modelling programme? For example, what do biologists, economists, industrial and financial planners, medical practitioners, etc., look for in a student’s mathematical modelling abilities? How does one establish whether a student has achieved these capabilities? 3.9. Technological impacts Many technological devices are available today and many of them are highly relevant for applications and modelling. In a broad sense these technologies include calculators, computers, Internet and all computational or graphical software as well as all kind of instruments for measuring, for performing experiments, for solving all kind of daily life problems, etc. These

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devices provide not only increased computational power but broaden the range of possibilities for approaches to teaching, learning and assessment. Moreover, the use of technology is in itself a key knowledge in today’s society. On the other hand, the use of calculators and computers may also bring inherent problems and risks. Issue 9: Technology can obviously provide support for well-structured mathematical problems that students will meet in their mathematical studies from the lower secondary level on. How should technology be used at different educational levels to effectively develop students’ modelling abilities and to enrich the students’ experience of openended mathematical situations in applications and modelling?

Examples of specific questions: – What implications does technology have for the range of applications and modelling problems that can be introduced? – What important aspects of applications and modelling are touched (or not touched) upon by the technological environment? – How is the culture of the classroom influenced by the presence of technological devices? Will button pressing compromise thinking and reflection or can these be enhanced by technology? – What evidence of successful or failed practice in teaching and learning applications and modelling has been documented as a direct consequence of the introduction of technology? – In what cases does technology facilitate the learning of applications and modelling? When may technology kidnap learning possibilities, e.g. by rendering a task trivial, when can it enrich them? – In which cases is technology a crucial need in modelling in the classroom? Are there circumstances (if any) where modelling processes can’t be developed without technology? – With respect to non-affluent countries: can applications and modelling be successfully done without any technology? – What are the implications of the availability of technology for the selection of assessment items and practices for use in contexts involving applications and modelling?

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4. C ALL FOR CONTRIBUTIONS TO THE STUDY

The ICMI Study on Applications and Modelling in Mathematics Education will consist of three components: an invited Study Conference, a Study Volume and a Study Website. The Study Conference will be held in Dortmund (Germany), February 13–17, 2004. The conference will be a working one where every participant will be expected to be active. As is the normal practice for ICMI studies, participation in the study conference is by invitation only, given on the basis of a submitted contribution, and is limited to approximately 75 people. The Study Volume, to be published after the conference in the ICMI Study Series, will be based on selected contributions and reports prepared for the conference, as well as on the outcomes of the conference. The Study Website, accessible also after the conference, will contain selected examples of good practice in applications and modelling. A report on the Study and its outcomes will be presented at the 10th International Congress on Mathematical Education to be held in Copenhagen in July 2004. The International Programme Committee (IPC) for the Study invites submission of contributions on specific questions, problems or issues related to this Discussion Document. Contributions, in the form of synopses of research papers, discussion papers or shorter responses, may address questions raised above, or questions that arise in response, or further issues relating to the theme of the Study. Submissions should not exceed 6 pages in length and should reach the Programme Chair at the address below (preferably by e-mail) no later than June 15, 2003, but earlier if possible. All submissions must be in English, the language of the conference. The contributions of those invited to the conference will be made available to other participants among the conference materials or on the conference website. However, an invitation to the conference does not imply that a formal presentation of the submitted contribution will be made during the conference. It is hoped that the conference will attract not only ‘experts’ but also some ‘newcomers’ to the field with interesting and refreshing ideas or promising work in progress. Unfortunately an invitation to participate in the conference does not imply a financial support from the organisers, and participants should finance their own attendance at the conference. Funds are being sought to provide partial support to enable participants from nonaffluent countries to attend the conference, but it is unlikely that more than a few such grants will be available.

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The members of the International Programme Committee for this Study are: – – – – – – – – – – – – – –

Werner Blum (University of Kassel, Germany), Chair of the IPC Claudi Alsina (University of Technology, Barcelona, Spain) Maria Salett Biembengut (University of Blumenau, Brazil) Nicolas Bouleau (École Nationale des Ponts et Chaussées, Marne-laVallée, France) Jere Confrey (University of Texas-Austin, USA) Peter Galbraith (University of Queensland, Brisbane, Australia) Toshikazu Ikeda (Yokohama National University, Japan) Thomas Lingefjärd (Gothenburg University, Sweden) Eric Muller (Brock University, St. Catharines, Canada) Mogens Niss (Roskilde University, Denmark) Lieven Verschaffel (University of Leuven, Belgium) Shangzhi Wang (Capital Normal University, Beijing, China) Bernard R. Hodgson (Université Laval, Québec, Canada), ex officio, representing the ICMI Executive Committee Hans-Wolfgang Henn (University of Dortmund, Germany), Chair of the Local Organising Committee.

For further information and submission of contributions, please contact the Chair of the IPC: Prof. Dr Werner Blum, University of Kassel, Fachbereich Mathematik/Informatik, D-34109 Kassel, Germany, Tel: +49 561 804 4623 (secr. –4620), Fax: +49 561 804 4318, e-mail: [email protected]

R EFERENCES We include some basic references (in English language) published on this topic in recent years, most of them generated by ICMI and ICTMA activities: Bell, M.: 1983, ‘Materials available worldwide for teaching applications of mathematics at the school level’, in Zweng, M. et al. (eds.), Proceedings of the Fourth International Congress on Mathematical Education, Birkhäuser, Boston, pp. 252–267. Berry, J. et al. (eds.): 1984, Teaching and Applying Mathematical Modelling, Ellis Horwood, Chichester.

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Berry, J. et al. (eds.): 1986, Mathematical Modelling Methodology, Models and Micros, Ellis Horwood, Chichester. Berry, J. et al. (eds.): 1987, Mathematical Modelling Courses, Ellis Horwood, Chichester. Blum, W. et al. (eds.): 1989, Applications and Modelling in Learning and Teaching Mathematics, Ellis Horwood, Chichester. Blum, W., Niss, M. and Huntley, I. (eds.): 1989, Modelling, Applications and Applied Problem Solving - Teaching Mathematics in a Real Context, Ellis Horwood, Chichester. Blum, W., Niss, M.: 1991, ‘Applied mathematical problem solving, modelling, applications, and links to other subjects – State, trends and issues in mathematics instruction,’ Educational Studies in Mathematics 22(1), 37–68. Boyce, W.E. (ed.): 1981, Case Studies in Mathematical Modeling, Pitman Ad. Pub., Boston. Breiteig, T., Huntley, I. and Kaiser-Meßmer, G. (eds.): 1993, Teaching and Learning Mathematics in Context, Ellis Horwood, Chichester. Burghes, D., Huntley, I. and McDonald, J.: 1982, Applying Mathematics – A Course in Mathematical Modelling, Ellis Horwood, Chichester. Burkhardt, H. (ed.): 1983, An International Review of Applications in School Mathematics, ERIC, Ohio. Burkhardt, H.: 1981, The Real World and Mathematics, Blackie and Son, Glasgow. Bushaw, D. et al. (eds.): 1980, A Sourcebook of Applications of School Mathematics, NCTM, Reston. Clements, R. et al. (eds.): 1988, Selected Papers on the Teaching of Mathematics as a Service Subject, Springer, Berlin/Heidelberg/New York. COMAP, The UMAP Journal, COMAP, Lexington. COMAP: 1997–1998, Mathematics: Modeling Our World, South-Western Ed. Pub., Cincinnati. deLange, J. et al. (eds.): 1993, Innovation in Maths Education by Modelling and Applications, Ellis Horwood, Chichester. deLange, J.: 1996, ‘Using and applying mathematics in education’, in Bishop, A. et al. (eds.), International Handbook of Mathematics Education V.1, Kluwer Acad. Pub., Dordrecht, pp. 49–97. Galbraith, P. and Clathworthy, N.: 1990, ‘Beyond standard models – Meeting the challenge of modelling’, Educational Studies in Mathematics 21(2), 137–163. Galbraith, P. et al. (eds.): 1998, Mathematical Modelling – Teaching and Assessment in a TechnologyRich World, Ellis Horwood, Chichester. Giordano, F.P., Weir, M.D. and Fox, W.P.: 1997, A First Course in Mathematical Modeling, Brooks, Pacific Grove. Houston, S.K. et al. (eds.): 1997, Teaching and Learning Mathematical Modelling, Albion Pub., Chichester. Howson, G. et al. (eds.): 1988, Mathematics as a Service Subject, Cambridge University Press, Cambridge. Huntley, I. and James, G. (eds.): 1990, Mathematical Modelling – A Source Book of Case Studies, Oxford University Press, Oxford. Klamkin, M.S. (ed.): 1987, Mathematical Modelling: Classroom Notes in Applied Mathematics, SIAM, Philadelphia. Lesh, R.A. and Doerr, H. (eds.): 2002, Beyond Constructivism: A Models and Modelling Perspective on Teaching, Learning, and Problem Solving in Mathematics Education, Lawrence Erlebaum, Mahwah.

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MAA and NCTM (eds.): 1980, A Sourcebook of Applications of School Mathematics, NCTM, Reston. Matos, J.F. et al. (2001.): Modelling and Mathematics Education: ICTMA-9, Ellis Horwood, Chichester. Niss, M.: 1987, ‘Applications and modelling in the mathematics curriculum – State and Trends’, International Journal for Mathematics Education in Science and Technology 18, 487–505. Niss, M.: 1992, ‘Applications and modelling in school mathematics – Directions for future development’, in Wirszup, I. and Streit, R. (eds.), Development in School Mathematics Education Around the World V.3, NCTM, Reston, pp. 346–361. Niss, M.: 2001, Issues and Problems of Research on the Teaching and Learning of Applications and Modelling, in Matos et al., loc. cit, pp. 72–88. Niss, M., Blum, W. and Huntley, I. (eds.): 1991, Teaching of Mathematical Modelling and Applications, Ellis Horwood, Chichester. OECD (ed.): 1999, Measuring Student Knowledge and Skills – A New Framework for Assessment, OECD, Paris. Pollak, H.O.: 1997, ‘Solving problems in the real world’, in Steen, L.A. (ed.), Why Nymbers Count: Quantitative Literacy for Tomorrow’s America, The College Board, New York, pp. 91–105. Pollak, H.O.: 1979, ‘The interaction between mathematics and other school subjects’, in UNESCO (ed.), New Trends in Mathematics Teaching IV, Paris, pp. 232–248. Pozzi, S., Noss, R. and Hoyles, C.: 1998, ‘Tools in practice, mathematics in use’, Educational Studies in Mathematics 36(2), 105–122. Sharron, S. (ed.): 1979, Applications in School Mathematics, NCTM Yearbook, NCTM, Reston. Sloyer, C., Blum, W. and Huntley, I. (eds.): 1995, Advances and Perspectives in the Teaching of Mathematical Modelling and Applications, Water Street Mathematics, Yorklyn. Stillman, G.: 1998, ‘Engagement with task context of applications tasks: Student performance and teacher beliefs’, Nordic Studies in Mathematics Education 6(3-4), 51–70. Stillman, G. and Galbraith, P.: 1998, ‘Applying mathematics with real world connections: Metacognitive characteristic of secondary students’, Educational Studies in Mathematics 36(2), 157–195. Swetz, F. and Hartzler, J. (eds.): 1991, Mathematical Modelling in the Secondary School Curriculum, NCTM, Reston. Verschaffel, L., Greer, B. and De Corte, E.: 2000, Making Sense of Word Problems, Swets&Zeitlinger, Lisse.

MICHÈLE ARTIGUE CHAIR OF THE ICMI AWARDS COMMITTEE

INFORMATION ABOUT THE ICMI AWARDS

The Executive Committee of the International Commission on Mathematical Instruction has decided to create two awards in mathematics education research, at its annual meeting in 2000: • the Hans Freudenthal Award, for a major program of research on mathematics education during the past 10 years, • the Felix Klein Award, for lifelong achievment in mathematics education research. These awards will consist of a certificate and a medal, and they will be accompanied by a citation. They should have a character similar to that of a university honorary degree, and they shall be given in each odd numbered year. At each ICME, the medals and certificates of the awards given after the previous ICME will be presented at the Opening Ceremony. The first recipients of the Freudenthal and Klein awards, will be known by the end of the year 2003. These awards will be formally presented at the opening ceremonies of ICME 10 in Copenhagen. An Award Committee (AC) of six persons shall select the awardees. Members of the AC are appointed by the President of ICMI, after consultation with the Executive Committee and with other scholars in the field. The terms of appointment is for 8 years and non-renewable, with three of the members being replaced each four years, at the time of the ICME’s. One of the three continuing members shall then also be named as committee chair. To initiate the process, a committee of 6 has been appointed in 2002, three of them with 8-year terms, the other three with 4year terms. Michele Artigue, professor at the university Paris 7 in France, and one of the vice-presidents of ICMI has accepted the task of chairing the first Award Committee, with a term of 4-years. The active members of the AC, except for its chair, shall not made be known. Only at the time when the terms of committee members expire shall their names be made public. The AC, once appointed, is completely autonomous. Its work and records are kept internal and confidential, except for the obvious process of soliciting advice and information from the professional community, which Educational Studies in Mathematics 51: 173–174, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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should be done by the committee chair. The committee has full authority to select the awardees. Its decision is final. Once made, that decision is to be reported, in confidence, to the ICMI-EC, via the President of ICMI. The AC is open to suggestions as regard future awardees. All such suggestions, which have to be carefully supported, must be sent by ordinary mail to the chair of the Committee, by the end of June 2003 (see the adress below). IREM, Université Paris 7, Case 7018, 2 Placve Jussieu, 75251 Paris Cedex 05, France

EDUCATIONAL STUDIES IN MATHEMATICS

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EDITOR-IN-CHIEF ANNA SIERPINSKA Department of Mathematics and Statistics, Concordia Universtity 7141 Sherbrooke St. West Montréal, Québec H4B1R6, Canada. E-mail: [email protected]

A IMS AND S COPE Educational Studies in Mathematics presents new ideas and developments which are considered to be of major importance to those working in the field of mathematical education. It seeks to reflect both the variety of research concerns within this field and the range of methods used to study them. It deals with didactical, methodological and pedagogical subjects rather than with specific programmes for teaching mathematics. All papers are strictly refereed and the emphasis is on high-level articles which are of more than local or national interest. All contributions to this journal are peer reviewed. O NLINE M ANUSCRIPT S UBMISSION Kluwer Academic Publishers now offers authors, editors and reviewers of Educational Studies in Mathematics the option of using our fully webenabled online manuscript submission and review system. To keep the review time as short as possible (no postal delays!), we encourage authors to submit manuscripts online to the journal’s editorial office. Our online manuscript submission and review system offers authors the option to track the progress of the review process of manuscripts in real time. Manuscripts should be submitted to www.editorialmanager.com/educ. The online manuscript submission and review system for Educational Studies in Mathematics offers easy and straightforward log-in and submission procedures. This system supports a wide range of submission file Educational Studies in Mathematics 51: 175–182, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Kagan, D.: 1990, ‘Ways of evaluating teacher cognition: Inferences concerning the Goldilocks Principle’, Review of Educational Research 60 (3), 419–469.

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Letter to the editor R.P. BURN

SOME COMMENTS ON ‘THE ROLE OF PROOF IN COMPREHENDING AND TEACHING ELEMENTARY LINEAR ALGEBRA’ BY F. UHLIG1

I write in response to the Editor’s invitation on page 335 of ESM 50.3 concerning Uhlig’s article on Linear Algebra. Within the conventions of 20th century university mathematics, Uhlig’s DLPTPC2 is the product – mathematics, and Uhlig’s WWHWT3 is the process of getting there. By grasping the distinction between these two modes of doing mathematics Uhlig has been addressing didactical problems and opening mathematics up to the guesswork and intuition of his students. The development of linear algebra during the eighteenth and early nineteenth centuries is largely concerned with determinants and quadratic forms, and this does not immediately suggest that a genetic development based on history would suit today’s students. Uhlig, however, makes a more subtle claim, pointing out that standards of rigour in all areas of mathematics sharpened around the end of the 19th century, with the development of axiom systems. He implies that the development from more intutitive to more rigorous mathematics is indeed a genetic development which one might expect to be reflected in today’s students, a finding which he usefully reports. In commenting on Uhlig’s course I am at a disadvantage in having taught linear algebra in a different context to intending teachers, and not to budding engineers. I am also at a disadvantage in not having access to many of the references he cites on page 342 against geometry. Aware that I may be walking blindfold through a minefield, I will hazard some remarks. For intending school teachers there can be no doubt that R2 and R3 are vector spaces which will be unavoidable throughout their professional lives, and for these spaces the relation between their geometric and algebraic properties can and should be explored. But when familiar with these two special cases, in which the notion of basis has emerged, one can extract common properties of them both in the form of vector space axioms. Then the axioms are satisfied by all manner of sets. Arithmetic progressions form a two-dimensional space. Linear functions of one variable form a Educational Studies in Mathematics 51: 183–184, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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two-dimensional space. Quadratic (and linear) functions of one variable form a three-dimensional space. Recognising the same structure (of R2 and R3 ) in unfamiliar garb is a surprising and satisfying experience. After achieving that, extension to n-dimensions or to infinite dimensional space (with sequences or functions) is almost natural. Taking this route, both R2 and R3 are concrete and generalisable. To rule out spatial intuition in vector spaces is a drastic step. Psychologically, students differ. There are those for whom linear equations are inert and geometry, dynamic by comparison. For them, rowreduction is a chore, and linear transformations of R2 a treasure chest. One last remark: there is enough evidence from the writings of distinguished mathematicians to know that no one ever outgrows the need for WWHWT explorations, testing the possibilities with special cases.

N OTES 1. Uhlig, F.: 2002, ‘The role of proof in comprehending and teaching elementary linear algebra’, ESM 50(3), 335–346. 2. Definition-Lemma-Proof-Theorem-Proof-Corollary. 3. What happens if? Why does it happen? How different cases occur? What is true here?

University of Exeter, U.K.

Commentary JEAN-LUC DORIER1 , ALINE ROBERT2 and MARC ROGALSKI2

SOME COMMENTS ON ‘THE ROLE OF PROOF IN COMPREHENDING AND TEACHING ELEMENTARY LINEAR ALGEBRA’ BY F. UHLIG1

ABSTRACT. In a recent issue of this journal (ESM 50.3) Frank D. Uhlig published a very interesting article about the question of proof in linear algebra. We have been doing research in the field of mathematics education about the teaching of linear algebra since the 1980s. In this paper, we want to underline the common points in Uhlig’s approach and some of our work. We also want to bring a new light on some of his ideas and give a perspective for further didactical development of Uhlig’s first experiments. KEY WORDS: higher education, history of mathematics, linear algebra, linear dependence, meta-lever, proof, reflective abstraction

The authors of this paper are researchers in mathematics education, who have been involved in several research projects on the teaching of linear algebra at the beginning of French university since the late 1980s. A book (Dorier, 2000)2 published by Kluwer in the series Mathematics Education Library gives an overview of their work as well as of other authors from various countries. We met Frank Uhlig during the annual meeting of the International Linear Algebra Society, in 1996. We have exchanged ideas since then and we have been very interested in his textbook (Uhlig, 2002a) which gives a very original approach to the teaching of linear algebra. The paper published recently in Educational Studies in Mathematics (Uhlig, 2002b) gives interesting clues to better appreciate the value of Uhlig’s approach. Our aim in this paper is to reflect on this approach with the experience from research in mathematics education and a long practice of teaching linear algebra and analyzing this teaching. We totally agree with Uhlig on the fact that the ‘classical’ teaching of linear algebra, which he characterises as the sophisticated ‘DefinitionLemma-Proof-Theorem-Proof-Corollary (DLPTPC) Approach’, leads to important difficulties in the students, which we have characterised as the ‘obstacle of formalism’. Not only students have difficulties in understanding proofs, but they are also overwhelmed by the number of new definitions and feel like they are landing on a new planet (an expression first used by Educational Studies in Mathematics 51: 185–191, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Hillel). Uhlig sees a general problem there, stemming from the lack of practice in proving in mathematics. In the French educational system, students may be better prepared for proof in mathematical teaching. Indeed, they encounter proof at the beginning of secondary school (age 13–14) in the context of elementary geometry. Using different characterisations of a parallelogram, the Pythagorean and Thales Theorems are important areas in the curriculum of geometry in French secondary schools. Moreover, the teaching of Calculus (which is more the beginning of real analysis) tends to be more formal and proof oriented at the end of secondary school and at the beginning of French science universities. For three years now, the teaching of elementary number theory is also part of the curriculum for students specialising in mathematics at the end of secondary schools. Therefore, it seems that, indeed, students are more accustomed to proving in mathematics classes in France than they are in the US. Nonetheless, they encounter similar problems when faced with their first courses in linear algebra. This means that the difficulties with proof and formalism in understanding linear algebra are content-specific. We have worked on this question for a long time. An extensive work in the history of mathematics (Dorier, 1995a and 2000, part I) supported an epistemological reflection that led to the idea of unifying and generalizing concept, first presented in (Robert and Robinet, 1996)3 . A unifying and generalizing concept (or theory) is characterised by the fact that it did not emerge essentially to solve a new type of problems in mathematics (like the derivative or the integral for instance). Its creation and its use by mathematicians were motivated rather by the necessity to unify and generalise methods, objects and tools, which had been independently developed in various fields. Therefore the formalism attached to a unifying and generalizing concept is constitutive of its existence and creation. This means that the formalism is not just a pure convenience of language or communication, but is an unavoidable part of the nature of the concept itself. In other words, formalism cannot be avoided when learning linear algebra; furthermore, learning linear algebra includes appreciating the value of formalism. This does not mean that unifying and generalizing concepts have no intuitive background. In fact they have several such backgrounds which result from an abstraction of the common characteristics of various objects of a less formal nature. Therefore, learning linear algebra requires that students look back to different previously learned fields of mathematics and step aside in order to have a reflective attitude towards what they already know. For instance, we share with Uhlig the idea that reflecting upon the solvability of systems of linear equations is an important starting point in order to access more formal ideas in linear algebra. A detailed description of a teaching project

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can be found in (Rogalski, 1996); (see also Dorier, Robert, Robinet and Rogalski, 1994 and 2000). We have especially worked on the concepts of linear dependence and rank. The main results of this work, including an interactive epistemological approach between the historical and the didactical contexts, can be found in (Dorier, 1998). We will try to summarise the results below. It is important that students have acquired a good technical level in solving systems of linear equations before we start to teach linear algebra. Several solving methods can be used. The determinants, which historically dominated the subject from 1750 up to the beginning of the twentieth century, are to be avoided since their technicality tends to mask basic ideas. Algorithmic approaches are to be favoured for their systematic nature. For cultural reasons, more than anything else, we use the Gaussian elimination, rather that the Row Echelon Form (REF), favoured by Uhlig, yet, we totally agree with him on this point. An historical analysis of several texts showed that the basic idea of linear dependence was not so easy to formalise, even by great mathematicians like Euler, who were first faced with the question of non-determination of n unknowns by any n equations when trying to solve what is known as Cramer’s paradox (Euler, 1750). Indeed, the idea of dependence between equations is not immediately attached to the idea of linear dependence. An equation is dependent on the other ones when it gives no new restrictions on the unknowns; this is what Dorier named the concept of ‘inclusive dependence’. This view on dependence of equations is totally consistent with the fact that the main goal was to solve equations, not to reflect on them as objects. Of course, the inclusive dependence is logically equivalent to the concept of linear dependence, but it is very different from a cognitive point of view. Historically, it prevented mathematicians from using the same concept of dependence on equations and on n-tuples and, therefore, of having a clear idea of duality, which is an essential step in order to totally build the concept of rank. It took nearly a century and a half before Frobenius reached this stage. A didactic study showed that sophomore students, before any teaching of linear algebra, hold, in their great majority, very similar pre-conceptions about dependence of equations. Indeed, as mathematicians of Euler’s time, they are uniquely concerned with solving equations. Therefore, an essential stage to prepare students to the formal concept of linear dependence is to make them step aside and reflect on the solving process and their actions when solving a system of linear equations. This can be done by reflecting on the reasons why a pivot is zero at the end of the Gaussian elimination. Here, we come to another similarity with Uhlig’s approach, even if we use

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different contexts. Indeed, to achieve the goal of having students reflect on their actions, we think that it is essential to ask them questions like: What happens if. . .? Why does it happen? How does it occur? What is true here?

This is what we call using the meta lever (Dorier, 1995b; Robert and Robinet 19964 ). Meta means that there is a reflection upon previously acquired knowledge. Lever means that this is something to be used at an appropriate time, in order to reach a higher level of knowledge. It is essential that the activities designed with the use of meta lever do not reduce to a discourse from the teacher followed by an activity by the students. The students have to be engaged in a mathematical task, which becomes problematic; the meta lever is then to be used at a stage where the students have started a reflection based on actions. It can be a discourse from the teacher, an historical text with questions, or just an unusual question, that is chosen in order to make students step aside and change their point of view. We have designed several such activities, for instance about the introduction of the axioms of vector space, or the use of a general method in problems of interpolation (Dorier, 1995b). In the teaching project that we experimented and improved several times, the progression of new material is deliberately slow, to allow for the necessary maturation of ideas in students. It also starts from several investigations and ‘meta lever activities’ in various contexts (linear equations, geometry, magic squares, Cartesian and parametric representations of geometrical objects,. . .). A first stage of formalism in Rn is attained, before formal vector spaces are introduced. However an essential step is the use of a unique letter to name the vectors in Rn and the differentiation between a matrix and the transformation associated via the original basis. This is an important difference with the usual North American curriculum, which allows to avoid the problem related to the notion of coordinate vectors analysed by Hillel and Sierpinska (1994) (see also Dorier, 2000, pp. 191–207). As can be seen, we share many ideas with Uhlig. Yet his approach is totally original. However, it raises new questions. Indeed, the extensive use made of the REF has the advantage of providing the students with a technical ground that is essential. The danger is that this becomes a ‘magic tool’ that hides some essential ideas. Let us examine for instance the third exercise that Uhlig proposes in section 4 of his article: For which vectors u in R4 are the three vectors u, u+e2 , and u+e3 in R4 linearly independent, where ei denotes the the ith unit vector in R4 ?

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Uhlig proposes a solution using his intuitive definition of linear dependence and challenges the reader to solve this question with the formal definition. Let us take up the challenge. The easiest is to start from the fact that the three vectors are linearly dependent. This is equivalent to the fact that there exist three scalars a, b, c not all equal to zero, such that au + b(u + e2 ) + c(u + e3 ) = 0. This is equivalent to (a+b+c)u = – be2 – ce3 (1). If a+b+c = 0 then be2 + ce3 = 0, which implies that b = c = 0 since e2 and e3 are unit vectors. This, together with a+b+c=0 implies that a = 0, and therefore the three scalars should be zero, which contradicts the assumption that the vectors are dependent. Therefore a+b+c =0 and one can divide (1) by a+b+c. This implies that the three vectors are linearly dependent iff there exists two scalars m and n such that u = me2 + ne3 , which means that u belongs to span (e2 , e3 ). We totally agree that this proof is complicated and requires quite a lot of sophisticated skills in logic, as well as in linear algebra. Yet, it gives a very different light on the result compared to the computational proof proposed by Uhlig. Moreover, it is very doubtful that the computational proof will prepare students for such a formal proof. They will be certain of the result, but they will still lack a certain level of comprehension that it involves. The question is to know whether we want our students to attain this level of comprehension? If yes, is it possible and how? Uhlig seems to assume that his method leads ‘naturally’ to a more formal stage. What formal stage does he have in mind? Is it possible to attain the level of the preceding formal proof? He claims that his approach provides an intuitive basis for linear algebra concepts. We would rather say that it gives a technical basis that allows to attain a large number of conceptual results. The question is to know how students can get free from this technique that governs most of their actions? It would be interesting to have more information on this point. Uhlig’s paper presents an original teaching proposal, which shares many ideas with our own research5 . A didactical challenge would be now to confront this proposal with an experimental didactical analysis in order to probe the effectiveness of its goals. Moreover, it seems important to evaluate the epistemological background vehiculated by the proposal. N OTES 1. Uhlig, F.: 2002, ‘The role of proof in comprehending and teaching elementary linear algebra’, ESM 50.3, 335–346. 2. See also Dorier and Sierpinska (2001) and Dorier (2002). 3. We only give here the most accessible reference but the ideas presented in this paper were first published around 1993.

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4. In spite of the dates of these papers, the ideas are originally Robert’s. 5. An issue that we have not discussed here concerns the relation with geometry. For recent ideas on this matter see (Gueuet-Chartier, in press).

R EFERENCES Dorier, J.-L.: 2002, ‘Teaching linear algebra at university’, in Li Tatsien (ed.), Proc. Int. Congr. Mathematician, Beijing 2002, August 20–28, Vol III (Invited Lectures), pp. 875– 884. Dorier, J.-L. and Sierpinska, A.: 2001, ‘Research into the teaching and learning of linear algebra’, in D. Holton et al. (eds.), The Teaching and Learning at University Level – An ICMI Study, Kluwer Academic Publishers, Dordrecht, pp. 253–271. Dorier, J.-L. (ed.): 2000, On the Teaching of Linear Algebra, Kluwer Academic Publishers, Dordrecht, 288 + xxii p. Dorier, J.-L., Robert, A., Robinet, J. and Rogalski, M.: 2000, ‘On a research program about the teaching and learning of linear algebra in first year of French science university’, International Journal of Mathematical Education in Sciences and Technology 31(1), 27– 35. Dorier, J.-L.: 1998, ‘The role of formalism in the teaching of the theory of vector spaces, Linear Algebra and its Applications 275-276, 141–160. Dorier, J.-L.: 1995a, ‘A general outline of the genesis of vector space theory’, Historia Mathematica 22(3), 227–261. Dorier, J.-L.: 1995b, ‘Meta level in the teaching of unifying and generalizing concepts in mathematics’, Educational Studies in Mathematics 29(2), 175–197. Dorier, J.-L., Robert, A., Robinet, J. and Rogalski, M.: 1994, ‘The teaching of linear algebra in first year of French science university’, in Proc. 18th Conf. Int. Group for the Psychology of Mathematics Education, Lisbon, 4(4), pp. 137–144. Euler, L.: 1750, ‘Sur une contradiction apparente dans la doctrine des lignes courbes’, Mémoires de l’Académie des Sciences de Berlin 4, 219–223, or in Opera omnia, (3 series – 57 vols.) Lausanne: Teubner – Orell Füssli – Turicini 1911–76(26), pp. 33–45. Gueudet-Chartier, G. (in press): ‘Using geometry to teach and learn linear algebra’, Research in Collegiate Mathematical Education, AMS, Providence, Rhode Island. Hillel, J. and Sierpinska, A.: 1994, ‘On one persistent mistake in linear algebra’, in Proc. 18th Int. Conf. on the Psychology of Mathematics Education, Lisbon, August 1994., Vol. III, pp. 65–72. Robert, A. and Robinet, J.: 1996, ‘Prise en compte du méta en didactique des mathématiques’, Recherches en Didactique des Mathématiques 16(2), 145–176. Rogalski, M.: 1996, ‘Teaching linear algebra: role and nature of knowledge in logic and set theory which deal with some linear problems’, in L. Puig et A. Guitierrez (eds.), Proc. XXth Int. Conf. for the Psychology of Mathematics Education, 4 vol., Valencia: Universidad, 4: pp. 211–218. Uhlig, F.: 2002a, Transform Linear Algebra, Prentice-Hall, Upper Saddle River, 504+xx p. Uhlig, F.: 2002b, ‘The role of proof in comprehending and teaching linear algebra’, Educational Studies in Mathematics 50.3, 335–346.

COMPREHENDING AND TEACHING ELEMENTARY LINEAR ALGEBRA 1

Equipe DDM – Laboratoire Leibniz, 46, Ave F. Viallet, 38 031 Grenoble, France E-mail: [email protected] 2

Equipe DIDIREM – IREM – Case 7018, Université Paris 7, 2, Place Jussieu, 75252 Paris Cedex 05, France E-mail: [email protected], [email protected]

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ABSTRACT. We report results from an analysis of responses to a written question in which high-attaining students in English schools, who formed part of a longitudinal nationwide survey on proof conceptions, were asked to assess the equivalence of two statements about elementary number theory, one a logical implication and the other its converse, to evaluate the truth of the statements and to justify their conclusions. We present an overview of responses at the end of Year 8 (age 13 years) and an analysis of the approaches taken, and follow this with an analysis of the data collected from students who answered the question again in Year 9 (age 14 years) in order to distinguish learning trajectories. From these analyses, we distinguished three strategies, empirical, focussed-empirical and focusseddeductive, that represent shifts in attention from an inductive to a deductive approach. We noted some progress from Year 8 to Year 9 in the use of the focussed strategies but this was modest at best. The most marked progress was in recognition of the logical necessity of a conclusion of an implication when the antecedent was assumed to be true. Finally we present some theoretical categories to capture different types of meanings students assign to logical implication and the rationale underpinning these meanings. The categories distinguish responses where a statement of logical implication is (or is not) interpreted as equivalent to its converse, where the antecedent and consequent are (or are not) seen as interchangeable, and where conclusions are (or are not) influenced by specific data. KEY WORDS: deduction, logic, logical implication, proof

I NTRODUCTION A fundamental objective of mathematics education must be to help students recognise and construct mathematical arguments, that is to engage in the process of proving. Hanna (2000) lists eight different functions of proof and proving, of which she regards verification and explanation as the most fundamental. When it comes to the mathematics classroom, Hanna places particular emphasis on explanation, which is surely right. However, verification and explanation clearly interact and our interest in this paper is on the former, and specifically the role in verification played by logical implication. Rodd (2000) argues that logical implication in the form of modus ponens reasoning (p⇒q, p so q), is one of the most basic structures for establishing a mathematical truth. Logical implication is also central to school mathematics, and to what Sowder and Harel (1998) call analytic proof schemes. Educational Studies in Mathematics 51: 193–223, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Deductive proofs of this type have the potential to be transparent, since “the validity of the conclusion flows from the proof itself, not from any external authority” (Hanna, 1995, p. 46). Yet it is well known in mathematics education that most school students do not find the deductive process straightforward and tend to use inductive reasoning1 to validate conjectures in mathematics rather than to prove them deductively (e.g., Bell, 1976; Van Dormolen, 1977; Balacheff, 1988). Even when students seem to understand the function of proof in the mathematics classroom (e.g., Hanna, 1989; de Villiers, 1990; Godino and Recio, 1997) and to recognise that proofs must be general, they still frequently fail to employ an accepted method of proving to convince themselves of the truth of a new conjecture, preferring instead to rely on pragmatic methods and more data (e.g., Fischbein, 1982; Vinner, 1983; Coe and Ruthven, 1994; Rodd, 2000; Simon, 2000). In the Fischbein study, for example, it was reported that only a minority of students judged that empirical checks would not increase their confidence in a proof that had already been accepted as valid and general. Fischbein argued that this seemingly contradictory behaviour was due to the fact that while students were being asked about a mathematical proof, their experience was mostly with empirical proof: “This (mathematical) way of thinking, knowing and proving, basically contradicts the practical adaptive way of knowing which is permanently in search of additional confirmation” (ibid, p. 17). How students learn to move between mathematical ways of proving and those that are rooted in everyday thinking is at the heart of our study. Since the 1980s in the U.K., there has been a shift of emphasis in the curriculum from geometry to simple number theory (see Hoyles, 1997). This shift has had two consequences. First, students have had few opportunities to engage with the deductive process in general and with the structure of a logical implication in particular. In the past when students were taught Euclidean proofs, they would have come across conditional relationships when learning about a proof and its converse, although it is unlikely that the formal connection between them would have been made explicit. Second, given that attention was largely paid to patterns in numbers rather than underlying mathematical relationships (see for example, Hewitt, 1992), many students have become unused to developing arguments based on structures rather than data. In a recent study of student proof conceptions (Healy and Hoyles, 2000), it was reported that, in line with previous research, most U.K. students were aware that a valid proof must be general. Additionally, when asked to assume a statement was true, and then to consider whether the statement would hold in a more restricted domain (e.g., for square numbers, when it

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was assumed to hold for whole numbers), most students did not feel that they needed more data, a finding which to some extent contrasts with that from the Fischbein study, mentioned earlier.2 Healy and Hoyles also found that this recognition of the general applicability and logical necessity of the conclusion of a theorem was a significant predictor of students’ overall performance on their proof test. Another strand of research in the area of proving in school mathematics has shed light on how the processes of explanation, justification and even logical necessity can be fostered by teachers (see for example Zack and Graves, 2001; Yackel and Cobb, 1996). Yackel (2001) in particular took Toulmin’s scheme (Toulmin, 1958) comprising conclusion, data, warrant and backing, as elaborated for mathematics education by Krummheuer (1995), to analyse interactive argumentation. Though a written question is far removed from the kind of interaction reported by Yackel, we also have adopted Toulmin’s scheme as part of our analysis. As background to the research reported here, we first briefly turn to review studies into students’ appreciation of logical relations and investigations of the logical properties of students’ arguments. BACKGROUND In many areas of mathematics education there has been considerable research into how students come to understand fundamental mathematical structures and relationships, for example function and proportion. By contrast, there has been rather little research into students’ understandings of the structure of logical relations, and in particular of logical implication. Research into children’s understanding of logical reasoning has largely been undertaken in the field of developmental psychology, originating with the seminal work of Inhelder and Piaget (1958), who used propositional calculus as a basis for their analysis of children’s responses: for example, they argued that understanding an implication p⇒q required an appreciation of its equivalence to the appropriate four combinations of truth values of the two propositions p and q, where p is the antecedent and q the consequent. In considering this formal structure of implication, it is important to recognise that the significance of combinations of truth values differs according to whether one is using an implication to draw conclusions (e.g., O’Brien, Shapiro and Reali, 1971), or trying to determine whether an implication is true (e.g., Wason, 1960). Studies into children’s understanding of implication have usually focussed on the former situation, and have generally found that children (and indeed adults) have difficulty in un-

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derstanding that implication includes a consideration of the case when the antecedent is false. A further, perhaps more pertinent, result from this corpus of research is that children tend to treat a conditional statement and its converse as equivalent, a phenomenon described as ‘child logic’ (O’Brien et al., 1971).3 The items used in our study are closer to Wason’s selection task (see for example, Wason and Shapiro, 1971) than to the items used by O’Brien, since we ask students to determine whether or not various given implications are true. However, structurally our items are simpler than Wason’s classic task as well as being in a different (and non-arbitrary) domain. Additionally, since the 1970s, evidence has accumulated that children rarely argue solely on the basis of universal formal laws of logic or domainindependent abstract rules. There are different interpretations as to why this might be the case: that children call up pragmatic reasoning structures derived from experience in context (Cheng and Holyoak, 1985), or that their arguments are derived from knowledge and the way it is structured (Ceci, 1990). Anderson, Chinn, Chang, Waggoner and Yi (1997) found that in naturally occurring arguments children tended to omit parts of the logic of their deductions, to be cryptic when mentioning the known or obvious, and elliptical when expressing their position, giving no more information than was necessary. Nonetheless, Anderson et al. concluded that the children’s arguments were logically complete: that is, the framework they used was not inconsistent with formal logic. Anderson et al’s analysis focussed on what they termed informal deduction. We follow this approach and, in contrast to Wason and to Inhelder and Piaget, we are not concerned with the strict requirements of formal logic. Rather we draw on an important distinction made for example by Mitchell (1962), between material implication, represented by “p⇒q”, which is part of propositional logic, and hypothetical proposition, represented by “if p, q” that “asserts only what is the case if its antecedent is realised” (Mitchell, 1962, p. 64).4 We claim that when studying reasoning in school mathematics, the latter (hypothetical proposition) is a more appropriate interpretation of logical implication than the former (material implication), since in school mathematics, students have to appreciate the consequence of an implication when the antecedent is taken to be true5 (see also Deloustal-Jorrand, 2002). There is some evidence from cross-sectional studies of logical thinking that the use of child logic decreases with age (O’Brien et al., 1971). Similarly, some researchers have postulated a developmental hierarchy of mathematical justification, “from inductive (empirical) reasoning toward deductive reasoning and toward a greater level of generality” (Simon,

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2000), though clear evidence for this seems to be lacking. More specifically, there has not been any systematic study of students’ understandings of logical implication in the context of school mathematics, or how these understandings might develop and change over time – two surprising gaps in the corpus of research knowledge given the importance of deduction in school mathematics. The study reported aims to throw light on both of these under-researched areas.

M ETHODOLOGY The findings reported in this paper arise from The Longitudinal Proof Project, a study that aims to describe students’ learning trajectories in mathematical reasoning over time. The particular focus of this paper is an investigation of the following three (interrelated) research questions concerning students’ meanings of logical implication where the reference knowledge is simple number theory. 1. How do students who are not taught about the structural meaning of logical implication determine whether a statement of logical implication is true or not and do their approaches change over time? 2. Are students aware of the general applicability and logical necessity of the consequent in a statement of logical implication if both the statement and the antecedent are true? 3. How do students conceptualise the relationship of logical implication and its converse and does this conceptualisation change over time? Sample and research instruments In the Longitudinal Proof Project, data are collected through annual surveying of students in the highest attaining class (or classes) of randomly selected schools within nine geographically diverse English regions. The survey instruments were specially devised written proof tests comprising a range of questions designed to probe different aspects of proving in the domains of number/algebra and geometry. The first written test (the Year 8 test) was administered to 2663 students in 63 schools in June 2000 when the students were approaching the end of Year 8 (age 13 years). The same students (with some inevitable dropout) were tested again in May to July of 2001 using a new instrument (the Year 9 test) that was designed to trace developments in mathematical reasoning as well as to build crosssectional profiles of student understandings at a particular age and place of schooling. The Year 9 test includes some questions that are identical

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to those in the Year 8 test, some that are slightly modified and some that are new. The Year 8 and Year 9 tests were both administered in parallel versions, A and B, with the same questions presented in a different order. The same students have been tested again in June 2002 with the similar aim of testing both understandings and development. All the questions in the proof tests were designed in collaboration with mathematics teachers in five design schools and extensively piloted in these schools. During the pilot stage of each proof test, a detailed set of codes was drawn up and validated, partly on the basis of theoretical distinctions in proving to which we wished to pay attention, such as generalisations in algebra, visual argument in geometry, and partly on the basis of the student responses. All the scripts were coded by one of the authors and one other coder and the consistency and reliability of the coding outcomes regularly checked.6 The code frequencies for both Year 8 and Year 9 responses have been analysed using descriptive statistics7 and a small number of students (and their teachers) interviewed to probe responses in more detail and to assist in our interpretation of the analyses. In this paper, we present analyses of students’ responses to one question, L1, involving logical implication, which was one of nine in the 50minute Year 8 proof test, and appeared again as one of ten in the 55-minute Year 9 test (with the names of Joe and Fred changed to Pam and Viv). A question like L1 would not be familiar to students in England, although they would have come across odd and even numbers. L1 has several parts that aim to find out how students conceive of the structure of logical implication, how they attempt to determine the truth of a statement that comprises a logical implication, and how the latter process might interact with the former. Analyses of cross-sectional responses to the question as a whole provide a snapshot of students’ reasoning at a given point in time. Additionally, analyses of the same students’ responses over time provide a picture of how reasoning develops, when implication is used as part of mathematical argumentation in classrooms but its structure not made explicit. The students are presented with two statements, an implication and its converse, the first of which is false and the other true (see Figure 1). Students are asked to decide whether the two statements are saying the same thing, to make a conclusion assuming one of the statements is true, and to evaluate the truth of each statement in turn. Regarding these evaluations, we note that students are asked to determine whether a given statement (which is expressed as an implication, p⇒q) is true. Given our earlier discussion of deductive reasoning in school mathematics, the crucial combinations under investigation are those where the antecedent is true, and the

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Figure 1. Question L1, which concerns a statement of logical implication and its converse.

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TABLE I Distribution of Year 8 responses to L1a (N = 2663) Item

Response code

Yr 8 responses Frequency % Number

L1a

code 10: Incorrect (Yes) code 31: Correct (Yes changed to No) code 32: Correct (No) code 9: Miscellaneous incorrect (including no response)

71 15 13

1897 387 355

1

24

consequent either true or false.8 It is anticipated that how students approach these evaluations will shed light on our other areas of interest, namely, the prior knowledge and the evidence generated within the question that students use to decide whether a statement and its converse are saying the same thing, and second, the extent to which students are able and willing to use a result to deduce a consequence.

R ESULTS We present the results, first through an analysis of Year 8 student responses to each part of question L1, and second through comparisons with the responses of some of the same students to the same question one year later. For the Year 8 analysis we report on the students (N = 2663) who took both the Yr 8 test and a ‘baseline maths test’ administered a few weeks earlier.9 Year 8 students’ responses to L1a We decided to ask students at the outset whether they thought Joe’s and Fred’s claims were the same (“. . . saying the same thing”). Clearly, responses to this might not be carefully thought out, and we expected that some students would change their answer after working through the later parts of the question, where there is further opportunity to think about the truth of each statement (Joe’s in L1c, Fred’s in L1d). In coding students’ answers to L1a, therefore, we distinguished between a straightforward ‘No’ and an answer that had clearly been changed from ‘Yes’ to ‘No’. The frequency distribution of responses is given in Table I. The table shows that 13% of students correctly stated from the outset that Joe’s and Fred’s statements were not saying the same thing, with a further 15% changing their answer from ‘Yes’ to ‘No’ at some stage. The

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TABLE II Distribution of Year 8 responses to L1b (N = 2663) Item Response code

L1b

code 10: Empirical (“Can’t be sure until you know what the numbers are”) code 30: Deduction (“Sum is even”) code 9: Miscellaneous incorrect (including no response)

Yr 8 responses Frequency % Number 47

1241

47

1249

6

173

vast majority of students, 71%, stated that the statements were saying the same thing. Year 8 students’ responses to L1b To investigate how far students are aware of the general applicability and logical necessity associated with a statement of logical implication, students were asked in L1b to assume that one of the statements (Fred’s) is true, and then to decide whether they could deduce a result from this assumption, or whether the result could still only be verified by induction (i.e., on the basis of further data).10 In L1b, students were told that the product of two whole numbers is 1271. This large and rather obscure number was chosen to discourage students from trying to find the possible values for the whole numbers (that is, 31 and 41, and 1 and 1271). Just under half the students in the sample (47%) chose the correct option (that the sum must be even) with another 47% choosing the empirical option (you can’t be sure until you know what the numbers are), as shown in Table II. Clearly, some students may have come to the correct conclusion by reference to properties of data rather than by a direct and general deduction (for example reasoning that two numbers whose product is 1271 must both be odd so their sum is even, or, indeed, that the numbers could be 1 and 1271 in which case the particular sum, 1272, is even). Nonetheless, it seems safe to assume that roughly half of the sample could play the mathematical game of supposing a statement is true, whether it is or not, and then making a correct deduction on that basis.

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TABLE III Distribution of Year 8 responses to L1c (Sum-even ⇒ Product-odd) (N = 2663) Item Response code

L1c

Yr 8 responses Frequency % Number

code 1: Correct or incorrect decision; invalid or no justification code 2: Correct or incorrect decision; flawed or incomplete justificationa code 3: Correct decision; valid justification, specific code 4: Correct decision; valid justification, general code 9: Miscellaneous incorrect (including terms not understood and no response)

26

684

22

589

28 8 16

753 216 420

a The code 2 responses were divided into the following sub codes:

code 21/23: Correct decision; flawed or incomplete justification (4%, 100 students). code 22: Wrong decision; justified by reference only to confirming cases (18%, 489 students).

Year 8 students’ responses to L1c: (Sum-even ⇒ Product-odd) L1c asks whether Joe’s statement is in fact true. The question can be answered correctly (namely that the statement is false) by means of a counter example. As Table III shows, 26% of the total Year 8 sample gave an invalid response or no justification (code 1) and another 22% gave responses that were only partially correct (code 2). Of these code 2 responses, most (18% of the total sample) stated that Joe’s statement was true and justified this with data that did indeed confirm the statement by picking a pair (or several pairs) of odd numbers (e.g., “Yes, Joe is right because 5 + 5 =10 (even), 5×5 = 25 (odd)”). Given the prevalence of these responses, they were given a separate code of 22. Thirty six percent of students correctly stated Joe’s statement was false and supported this by reference to the existence of a counter example. Of this 36%, most (28% of the total sample) gave specific counter examples (code 3), and usually just one (e.g., “No, e.g., 2 + 4 = 6, but 2×4 = 8 which is even, so they are both even”). Some students however (8% of the total sample) described the counter example in general terms (code 4), as consisting of even numbers (e.g., “No: even + even = even, but even × even = even also”). A general explanation of this sort is interesting for two reasons. First, it represents a shift from simply looking at data to considering underlying structure: it goes beyond showing that the statement is false, though this is all that the item requires, to giving some insight into why it is false. Second, a general counter example might have been found by

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TABLE IV Distribution of Year 8 responses to L1d (Product-odd ⇒ Sum-even) (N = 2663) Item Response code

L1d

Yr 8 responses Frequency % Number

code 1: Correct or incorrect decision; no valid justification code 2: Correct decision; incomplete justification, empiricalb code 4: Correct decision; valid justification, general code 9: Miscellaneous incorrect (including terms not understood and no response)

51

1353

24

643

9 16

242 422

b The code 2 responses were divided into the following sub codes:

code 21: Correct decision; confirmation by one empirical example (15%, 391 students). code 22: Correct decision; confirmation by several empirical examples (6%, 163 students). code 23: Correct decision; confirmation by one or several empirical examples and awareness that this is not conclusive (0%, 5 students). code 24: Correct decision; crucial experiment (confirmation by one or several empirical examples with at least one starting number greater than 10) (3%, 84 students).

deduction rather induction, by considering the mathematical relationships involved in the statement. Year 8 students’ responses to L1d: (Product-odd ⇒ Sum-even) L1d asks about the truth of Fred’s statement. In contrast to L1c, this can only be answered decisively by a deductive argument, for example, “If the product is odd, then the two numbers have to be odd and so the sum will always be even”, which would be accepted as appropriate for students of this age.11 Table IV shows that 24% of students correctly stated that Fred’s statement was true and supported this with correct but only empirical examples (code 2). Such responses usually again consisted of just one example (e.g., “Yes, 3×3 = 9, product odd; 3 + 3 = 6, sum even”). A further 9% of students supported their correct evaluation of Fred’s statement with a general description of the starting numbers, namely that they must both be odd (code 4). Such explanations were often cryptic so while all code 4 responses moved beyond reference to specific odd starting numbers, it was frequently difficult to distinguish responses that expressed the necessity of the starting numbers being odd and of the sum hence being even (i.e. a deductive approach), from responses where the need for odd numbers might have been the result of an inductive generalisation: (e.g., “I could only find odd

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numbers that made the product odd and sum even”). Finally some students were clearly influenced by their answer to L1c and wrote ‘same as Joe’, (11%, 294 students). Thus care has to be taken in comparing response frequencies between L1c and L1d as some frequencies might well have been different if the order had been reversed. Comparison of Year 8 and Year 9 students’ responses As mentioned earlier, the same question (with the names of Joe and Fred changed to Pam and Viv) was used again in the Year 9 survey administered in June 2001. Table V shows the code frequencies in Year 8 and Year 9, and their differences, for students’ responses to each part of question L1. (These frequencies are based on a smaller number of students (N = 1078) than the frequencies for Tables I to IV (N = 2663).12 For each item in Table V, the codes are ordered from 1 to 4, and it can be argued that codes with a higher leading digit represent a higher level of response (apart from code 9). On this basis, there is discernible but modest progress from Year 8 to Year 9 in students’ responses to each item; fewer students made an incorrect response, more students gave valid justifications and more students recognised that the implication and its converse presented in the question were not equivalent. The clearest progress can be seen in L1b, where there is a net migration of nearly 20% of the total sample of students from the incorrect code 10 to the correct code 30 response. Progress is less marked for the other items, being well under 10% in each case. Thus, for example, in item L1a there is a net migration of about 7% of students from the code 10 response to a code 31 or 32 response. There is a similar net percentage move from code 1 or 2 responses to code 3 or 4 responses in L1c and from code 1 responses to code 2 or 4 responses in L1d. In the case of L1a, though the net percentage progress was small, it is interesting to note that despite the complexity of the question (and its written test nature) a substantial minority of students (16% in Year 8, 19% in Year 9, N = 1078) were willing to re-assess their initial evaluation on the basis of their own proofs. Interestingly, these students who changed their answer from ‘Yes’ to ‘No’ on the basis of evidence achieved a somewhat higher mean score on the other numerical/algebraic items in the survey and a higher baseline maths test score than those who stayed with ‘Yes’ or ‘No’ from the outset. However, over 60% of students in each year still maintained that the two statements were ‘saying the same thing’. Since the study is longitudinal, as well as considering net progress, we are able to examine the percentages of actual students who progressed, regressed or gave the same kind of response to each item from Year 8 to

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TABLE V Changes in response frequencies for L1a, L1b, L1c and L1d from Year 8 to Year 9 (N = 1078) Item Code

L1a

L1b

L1c

L1d

code 10: Incorrect (Yes) code 31: Correct (Yes changed to No) code 32: Correct (No) code 9: Miscellaneous incorrect (including no response) code 10: Incorrect, empirical (“Can’t be sure”) code 30: Correct, deduction (“Sum is even”) code 9: Miscellaneous incorrect (including no response) code 1: Correct or incorrect decision; invalid or no justification code 2: Correct or incorrect decision; flawed or incomplete justification code 21/23: Correct decision; flawed justification code 22: Wrong decision; flawed justification code 3: Correct decision; valid justification, specific code 4: Correct decision; valid justification, general code 9: Miscellaneous incorrect (including terms not understood and no response) code 1: Correct or incorrect decision; no valid justification code 2: Correct decision; incomplete justification, empirical code 21: Correct decision; one empirical example code 22: Correct decision; several empirical examples code 23: Correct decision; aware examples not conclusive code 24: Correct decision; crucial experiment code 4: Correct decision; valid justification, general code 9: Miscellaneous incorrect (including terms not understood and no response)

c Frequencies are rounded to the nearest integer.

Frequency (%) Yr 8 Yr 9 Yr9–Yr8c 69 16 14 1

62 19 18 1

–7 3 4 0

42

23

–19

53

72

19

5

5

0

25

20

–5

21

19

–2

3 18 32 9 13

3 16 37 11 12

0 –1 5 2 –1

51

45

–5

24

27

3

14 7 0

13 11 0

–2 4 0

3 10

3 14

0 4

15

13

–1

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TABLE VI Percentage of total sample of students whose responses progress, regress or stay the same for items L1a, L1b, L1c and L1d from Year 8 to Year 9 (N = 1078) Item Frequency (% of total sample) Progress from Regress from Stay the same Code 9 response in Yr8 Yr8 to Yr9 Yr8 to Yr9 in Yr8 and Yr9 and/or Yr9 L1a L1b L1c L1d

25 26 28 22

16 9 20 15

56 54 29 37

2 11 23 26

Year 9. This is shown in Table VI, (where, as before, progress is defined in terms of the order of the leading digit, except for 9). Responses to L1b again stand out as indicated in Table VI by the difference in the Progress and Regress frequencies, which is higher for L1b than the other parts of the question. Additionally, if one considers the ratio of these frequencies, rather than the difference, this is also far greater for L1b (at roughly 3:1 compared to roughly 3:2 for each of the other items).

D ISCUSSION OF RESULTS In this section, we distinguish different patterns in student responses to separate parts of question L1 and attempt to interpret these patterns in terms of student strategies and meanings. We are also interested in the interactions between student responses to the various parts of question L1, and set out to trace interactions that led to students’ eventual evaluations as to whether an implication and its converse are equivalent. However we should note from the outset that inconsistency in students’ responses was not uncommon: for example, a student might appear to recognise the importance of satisfying the antecedent in a logical implication in answer to one part of the question but not in answer to another part. These inconsistencies led us to wonder how far students’ justifications of their conclusions (either in writing in this study or verbally during an interactive episode as reported in other studies) can be assumed to be based on an appreciation that if a justification is to be mathematical, it must be applied consistently. Further analysis of the strategies used by students reveal however that apparent inconsistent responses may sometimes be the result of a consistent

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use of strategies that happen to be limited in their effectiveness, as we discuss later. We begin by presenting what we think is a useful first approximation of the strategies students use to solve L1c and L1d (ignoring possible interactions between these parts) and how use of the strategies changed over time. Second, on the basis of responses to L1b, we discuss students’ awareness of the general applicability and logical necessity of a statement assumed to be true. Finally we analyse students’ responses to question L1 as a whole using a version of Toulmin’s scheme, in order to come up with a typology of students’ meanings for logical implication and the rationale underlying these meanings.

Student strategies to determine the truth of a statement of logical implication We present first a model of student attempts to determine the truth or otherwise of Joe’s and Fred’s statements. The model posits three types of strategy, which we call empirical (X), focussed-empirical (Y) and focusseddeductive (Z). The first of these is the least likely to produce a valid conclusion. Strategy X. In the empirical strategy, students start by more or less randomly choosing starting numbers, which are not restricted to those that fit the antecedent in the given statement, to generate data about sums and products. Subsequently, they try to relate the data to the statement they are trying to evaluate. Students are likely to use specific starting numbers, although some might reduce the amount of data by treating the numbers in a more general way, as simply ‘odd’ or ‘even’, and they might then also adopt an exhaustive approach for combining odds and evens. Both these additional approaches are likely to improve the chances of using the empirical strategy successfully. Strategy Y. In the focussed-empirical strategy, students start by only selecting starting numbers that fit the antecedent and then go on to produce sums or products in order to determine the truth-value of the consequent and hence of the statement as a whole. Again, some students might also adopt a general approach and perhaps an exhaustive approach as well. Strategy Z. In the focussed-deductive strategy, students start by asking themselves what types of starting number would fit the antecedent i.e., they use their insight or knowledge about the first proposition to deduce the starting numbers or starting number types. They then go on to produce sums or products in order to determine the truth-value of the consequent and hence of the statement as a whole, as with strategy Y. They then

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seek to explain their conclusion by reference to the original number type. Effectively, this strategy subsumes a general and exhaustive approach. The distinction between Y and Z is subtle and not always discernible from students’ scripts. Nonetheless, the distinction is useful from a theoretical perspective for two reasons. First, it denotes a change of viewpoint, from an empirical, data-driven approach to an approach that is deductive and based on structural relationships, i.e., to the kind of approach that is involved in mathematical proof. Second, the distinction brings out an important difference in how statements that turn out to be true, such as Fred’s, can be evaluated: strategies Y and Z, used accurately, both lead to the correct conclusion, that Fred’s statement is true, but whereas in Z it is necessarily true, this is not the case in Y – at least, not until other kinds of starting numbers have been exhaustively eliminated (i.e., odd-even and even-even pairs). In strategy X, the likelihood of coming up with a correct and valid conclusion will depend, at least in part, on the data that students happen to generate. Students using strategy X might well choose an odd and an even starting number (say 2 and 3), which produces data that is irrelevant, since it does not fit the antecedent for Joe (2+3 is not even) or for Fred (2×3 is not odd), but from which they might well be tempted to draw a conclusion which, even if it turns out to be correct, can not be valid. With Y and Z, on the other hand, the data is more focussed and under the students’ control. It is tempting to see the three strategies as ordered, both in terms of effectiveness and in terms of shifts in thinking, from an inductive approach to one that is deductive and concerned with structure that is appropriate for proof. Thus, for example, we noted a tendency for students who generated a lot of data (and who thus perhaps were using strategy X) to make wrong evaluations, particularly when the data did not fit the antecedent. However, this view has to be treated with caution, especially as the approach of treating the starting numbers only in terms of their odd/even property and the use of an exhaustive approach for generating data might be present. Other influences might also apply. For example, some students found data that fitted the antecedent in the implication (and thus perhaps used strategy Y), but took p⇒q to be true even though they had other data that would serve as counter examples (perhaps in the belief that ‘a statement is true if it is sometimes true’ by analogy with ‘a statement is false if it is sometimes false’). Many students’ produced empirical explanations, that is they involved one (or sometimes more) specific numerical examples rather than just referring to odds and evens, and often these comprised only small starting numbers (e.g., 2), and commonly identical starting numbers (i.e., both 2).

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Such numbers are far removed from numbers that would provide a ‘crucial experiment’ (Balacheff, 1988). However, it is possible that for some students these numbers served as generic examples. An obvious appeal is their economy, in terms of the arithmetic involved but also in terms of choice: having chosen a number of a certain type (for example, one that is even), the simplest way of choosing a second number of this type is to choose the same number again! In as much as Strategies Y and Z seem more likely to be successful than strategy X, it is possible that some of the (albeit modest) net progress (of about 10%) on L1c and L1d from Year 8 to Year 9 is due to a shift from using strategy X to using strategies Y or Z. Distinguishing these strategies also provides a possible explanation for the apparent inconsistency of many students’ responses to parts L1c and L1d, within a given year or between years. Thus, for strategies X and Y in particular, some inconsistencies could be due not to students switching strategies but to the fact that finding the crucial combinations for evaluating Joe’s and Fred’s statements is to some extent a matter of chance. This could also help to explain why a relatively high proportion of students seem to regress on L1c and L1d, compared to L1b. Students’ appreciation of the logical necessity of an implication This issue was investigated largely through responses to question L1b. Here, students are asked to suppose that Fred’s statement (Product odd ⇒ sum even) is true and whether, on that basis, it is then possible to draw a conclusion about the sum of two whole numbers whose product is 1271, or whether the value of the numbers has first to be determined. In contrast to Fischbein’s study reviewed earlier and in a similar way to the earlier Healy and Hoyles study, students were not given a proof of the statement and nor did they have to consider whether the statement was actually true. The idea was to test whether students were able to free themselves from the empirical and consider the implications of a statement at a structural level. In the event, a high proportion of our students operated successfully at this general level on item L1b, a proportion far higher than for those who seemed to operate at a general level on L1c and, in particular, L1d, where such an approach was needed for a conclusive argument. Thus responses to L1b provide useful evidence that many students (at least in our high-attaining sample) can appreciate the logical necessity of an implication when the antecedent is true and reason at this level in certain circumstances. Also, since students’ performance on L1b was found to be a significant predictor in the multilevel analysis of their score on the proof

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test as a whole, which replicates a finding from the Healy and Hoyles (2000) study, it would seem that considering the implications of a statement at a purely hypothetical level is an important component of learning to cope with mathematical proving.

Students’ meanings of logical implication and its converse In contrast to the preceding sections, our interest here is not so much on the way students evaluate the individual statements (Joe’s and Fred’s), but on how they decide on whether the two statements are ‘saying the same thing’. We present a typology of student responses to L1 as a whole, which attempts to capture the different ways students conceptualise the relationship of logical implication and its converse and the rationale underpinning these different meanings, in particular how far they are influenced (or not) by data. To come up with different types of argument, we use Toulmin’s scheme comprising conclusion, data, warrant and backing. The ‘conclusion’ is the claim about whether the statement p⇒q is saying the same thing as the statement q⇒p; the ‘data’, which provide a foundation for the claim, is the information about the form of the two statements p⇒q or q⇒p in this particular case or about their truth values for the specific propositions stated by Joe and Fred; the ‘warrant’ is a general principle by which the student seeks to justify why the data support the conclusion, and which is given legitimacy by the ‘backing’, which might involve further data and warrants. We distinguish four theoretical categories of student meanings for logical implication, referred to as Types A to D, which capture students’ conclusions about the equivalence (or not) of logical implication and its converse and the basis for their conclusions after answering the question as a whole.13 The typology was devised partly from theory: one might expect some students to equate a conditional statement with its converse whilst other students might, from the outset, regard them as different (perhaps, in this particular context, through an awareness of the structure of odd and even numbers, or perhaps more generally on the basis of a ‘container’ scheme as suggested by Rodd, 2000). The different categories in this typology were also however derived from close inspection of the students’ scripts and the analysis we have presented earlier.14 We describe each category or type by reference to the basic schematic diagram used by Krummheuer, 1995, based on Toulmin’s categories (see Figure 2). Since the backing will not call upon formal logic, the schematic diagram is repeated for some types to indicate that the backing is the conclusion from more data.

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Figure 2. Schematic diagram for analysing arguments.

In all that follows, proposition p is ‘sum is even’ and proposition q is ‘product is odd’.

Type A: Logical implication and its converse are the same (from the outset without reference to data) Type A responses make no reference to data concerning the truth values of antecedent and consequent, and simply regard antecedent and consequent as interchangeable. A typical Type A response profile can be summarised as follows. Students answer ‘Yes’ to L1a, and use as initial data the fact that in statements p⇒q and q⇒p, the particular propositions p and q are ‘just the other way round’. Students justify this with the general warrant that reversing the order of any propositions makes no difference to a statement of logical implication, and so p⇒q is the same as q⇒p for all p and q. Backing is given to the warrant by the claim that the particular statements (Joe’s and Fred’s), are either both true or both false. The crucial characteristic of a Type A response is that students do not make proper use of specific data to come to their conclusion; that is, they do not test the truth values of p⇒q and q⇒p independently, to demonstrate their equivalence. Rather, they test the truth value of L1c by reference to pairs of numbers (data for this statement) but then simply announce that the truth value of L1d is the same without reference to numbers.

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Figure 3. Schematic diagram for a Type A response.

An example from student interviews illustrates a typical Type A response. Student S, in her written response to the Year 8 test, had decided that Joe’s and Fred’s statements were saying the same thing, and she presented evidence that Joe’s statement was false. She then simply stated that Fred’s statement was false also. S was asked why she had written that the statements were saying the same thing: S:

Because Joe says that if the sum of two whole numbers is even, then the product is odd and Fred is saying the same thing, but he’s saying that if the product of two whole numbers is odd, then the sum is even. So I think that he’s basically saying that they’re showing that Joe works out sum of the numbers first and Fred works out the product first, but basically they think the same thing.

The schematic diagram summarising Type A responses, in the case when both statements are evaluated as true, is given in Figure 3. Type B: Logical implication and its converse are the same by reference to data Type B responses refer to data concerning the truth values of antecedent and consequent, while seeing antecedent and consequent as interchangeable.

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A typical Type B response profile can be summarised as follows. Students answer ‘Yes’ to L1a, and use as initial data the fact that in statements p⇒q and q⇒p, the propositions p and q are ‘just the other way round’. Students then use data about whether the particular statements p⇒q and q⇒p are true (that is, Joe’s and Fred’s statements) to back up their justification for the general equivalence of the two statements. In order to decide about the truth values of the statements p⇒q and q⇒p, the same pairs of numbers are used to produce the data, indicating that the antecedent and the consequent are seen as interchangeable. The clearest case of a Type B response is where the same pair of even numbers is used for both statements to show that one proposition (p) is true and the other (q) is false, which is taken to mean that both statements are false, thus backing the warrant asserting their general equivalence. We illustrate this response by an extract from an interview with student T. T felt the statements were saying the same thing and that the statements were both false since she was “. . .thinking just of the words. . . what they’re actually saying and whether it’s the same thing in English terms”. To explain this further, she switched to numbers: T:

. . .there is no way if you are talking in maths terms of deciding whether 4 times 2 caused (our emphasis) 4 plus 2 to be what it is or vice versa. . . for this particular thing there’s no distinction because 4 plus 2 is going to equal 6 no matter what 4 times 2 is . . . so um which ever way you put these round doesn’t matter because they don’t actually cause (our emphasis) each other to occur the way that they do. . .

T was expressing quite explicitly that she was seeing the propositions as interchangeable, i.e. one is not contingent on the other. This is illustrated in the schematic diagram in Figure 4. Another case that clearly fits Type B is when propositions p and q are both declared false, by using one odd and one even starting number, and where this in turn is used as evidence that both statements are false, and therefore equivalent. Such a response exhibits rather little understanding of how to evaluate the truth of an implication in school mathematics, since for neither Joe’s nor Fred’s statement is the antecedent true. Where students responded with the same pair of odd numbers, to produce data to show that both propositions and therefore both statements were true, it is not possible to decide whether they are distinguishing antecedent from consequent, and therefore whether the response is truly of Type B, though it fits the profile.

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Figure 4. Schematic Diagram for a Type B response.

Type C: Logical implication and its converse are not equivalent by reference to data Type C responses regard logical implication and its converse as not equivalent after testing with data the truth values of each specific statement. A typical Type C response profile can be summarised as follows. Students answer ‘Yes’ to L1a initially, but this is changed to ‘No’ after finding that L1c is false and L1d is true. Thus in Type C responses, it is asserted that a statement p⇒q is not the same as a statement q⇒p because p⇒q is false (since in this case there are data that make p true but q false), and q⇒p is true (since there are data that make q true and p true or, more powerfully, because there are also no data that make q true but p false). Thus, students whose responses fit Type C start with what appears to be a Type A or Type B response and then, after selecting appropriate pairs of numbers to find the truth values of L1c and L1d independently (and thus obtaining False, True), they change their answer to L1a from ‘Yes’ to ‘No’. We illustrate this response by a further extract from the interview with student S. Following her Type A response, the interviewer asked S to think about Fred’s statement without referring to Joe’s, as she had done in her

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Figure 5. Schematic Diagram for a Type C response.

written work. After first re-asserting that it would be false, S started to test this conjecture, and as she did so she became uncertain: S:

I think it’s probably false. Because I think if you have 10 and 8 the product would be – oh hold on, they’d be even (our emphasis). If you have . . . maybe he is right, I don’t know! I’m not sure about it now. . ..

Later, having been asked why 10 and 8 could not be used for Fred’s statement, S gave a reply that suggests that she was now beginning to see Fred’s statement as a hypothetical proposition, in that data that did not satisfy the antecedent could not be used: “Because then the product would be even, and it’s supposed to be odd”. S was now convinced that Fred’s statement was true, and having already shown that Joe’s statement was false, S changed her answer to L1a to ‘No’. This type of response is summarised by the schematic diagram shown in Figure 5.

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Type D: Logical implication and its converse are simply not equivalent In Type D responses, logical implication and its converse are seen as not equivalent; data concerning the truth values of antecedent and consequent are used to confirm rather than justify this conclusion. A typical Type D response can be summarised as follows. Students answer ‘No’ to L1a. They assert that a statement p⇒q is not the same as statement q⇒p, because the order matters, and they confirm this with subsequent data which shows that p⇒q is false and q⇒p is true. Not surprisingly, students are unlikely to give very clear reasons for their assertion of the non-equivalence. They just somehow ‘feel’ it is the case, as illustrated in the following extract from an interview with student W: W: I: W: I: W: I: W: W: W:

I don’t think they’re saying the same thing. Could you say why you don’t think they’re saying the same thing? Because, that’s not a simple reversal. Adding up and timesing is different. It depends what different numbers you have. How do you know it’s not a simple reversal? Because, there are different ways of doing things. Right. . . If you times something by one then you get the same but if you add a one then it makes a difference. . . Right. . . . . . And so, it changes and also divide and times would be a pair and I’d have to think about that more if it was something like that but because there are different ways, processes of doing something.

Thus W’s conviction that the two statements were not the same, though it might have been based on rather nebulous analogies with other mathematical experiences, did not rely on specific data. In Type D responses, students distinguish antecedent from consequent both at a general level and in their choice of pairs of numbers to determine (or demonstrate) the truth values of Joe’s and Fred’s statements. The schematic diagram summarising Type D responses is given in Figure 6.

C ONCLUDING REMARKS We start by recognising the limitations of findings that attempt to describe student understandings based on their response to written tests that can only give a partial view. Nonetheless we found some quite marked patterns and trends in the responses, which gives us some confidence in our conclusions. Additionally, the follow-up interviews with selected students

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Figure 6. Schematic Diagram for a Type D response.

lend support to our theoretical categories, although the interviews also showed how students can change their responses during interactions with a researcher. Overall, the research has shown that even high-attaining students often fail to appreciate how data can properly be used to support a conclusion as to whether p⇒q is true or not, and there is only modest progress from age 13 to age 14.15 Thus our study provides some but rather limited support to the assumption that students move from empirical to more deductive approaches with increasing maturity. It does however indicate the complexity underlying learning to prove deductively in mathematics and that progress is not likely therefore to be smooth and trouble free. In comparing responses across different parts of question L1, one striking feature is that the proportion of students who seemed to call upon general deductive reasoning is far higher in L1b, that is in recognition of the logical necessity of a conclusion of an implication when the antecedent was assumed to be true, than was evident in parts L1c and L1d. Additionally there was most marked and sustained progress in response to this item, suggesting that the higher level responses to L1b are generally more ‘secure’ than for the other three items; or put another way, there is perhaps a greater element of chance and uncertainty in some of the higher level responses to L1a, L1c and L1d.

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A more detailed examination of individual students’ scripts served to illustrate how students go about determining the truth of a statement of logical implication in this domain, and we distinguished three strategies from our analysis; empirical, focussed-empirical, and focussed-deductive. The first two strategies in particular may help to explain the inconsistency in students’ responses to L1c and L1d within and between years 8 and 9, since even if students use the strategies consistently, their success will depend on the particular data they happen to choose. The first strategy is likely to be the least successful, though the three strategies are not strictly hierarchical as they may (or may not) be used in conjunction with generic, general and exhaustive approaches. Our study identified a range of meanings students assign to the conditional relationship and its converse. Most assert that a conditional statement and its converse are equivalent and do not test this assertion against evidence of the truth values of the statements, and there is little if any progress over time. This argument could of course suggest superficial engagement with the question and an absence of reflection. However, it could also indicate a rather deep appreciation of mathematical structure in that a structural equivalence is declared so that there is no need to present further data to test it. By more detailed examination of the students’ responses to the question under study as whole, we distinguished four theoretical categories where logical implication is (or is not) interpreted as equivalent to its converse, where the antecedent and consequent are (or are not) seen as interchangeable, and where conclusions are (or are not) influenced by specific data. This analysis opens up questions as to the meanings students assign to their ‘reasons’ or verifications. It may not just be that reasons are not general, or not concerned with structure – they might simply be conjunctions of evidence. This calls into question what students mean when they present even a correct counter example. Is it in fact showing a recognition that the statement is not true or that the statement is sometimes not true? As we have noted, the majority of written explanations were limited to empirical reasoning but in considering empirical reasoning from this different vantage point, it is evident that many students used data in ways that did not distinguish the antecedent from the consequent, or, put another way, the students did not appreciate the temporal nature of if-then in the present context.16 One might speculate that students who recognise that the antecedent and consequent are not interchangeable would be able to cope with the general deduction required in part L1b. It is also likely that they would be able to answer L1c competently using a counter example obtained by a focussed-empirical or focussed-deductive approach and that

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they would be able to assess the truth value of L1d if not deductively then by finding supporting empirical evidence. It is also apparent that students who adopted only empirical reasoning did not always realise that it was not ‘the mathematical game’ to choose only confirming examples. This tendency, to want only to confirm a hypothesis, has long been recognised in experimental studies (Wason, 1960) and noted by Fischbein (1982) as a pragmatic and intuitive approach to proving. This research has led us to question the value of using formal logic as a model for studying deduction in secondary school mathematics, that is to follow the type of analysis used by Inhelder and Piaget (1958) in their study of hypothetic-deductive reasoning among adolescents or by Wason in his selection task. We have argued that in the context of school mathematics, logical implication is more usefully seen as being primarily concerned with situations in which the antecedent is true (rather than material implication where one is equally concerned with the antecedent being false). Related to this, we suggest that students are more likely to be successful in determining the truth of an implication if they use a focussedempirical or focussed-deductive approach on data that fit the antecedent than if they start by generating data ‘randomly’ and then attempt to assess the truth of the statement. We have shown some general, but rather modest, progress in responses as students moved on a year. At the same time, we can point to remarkable shifts in reasoning in some individuals. Why these shifts occurred is almost impossible to say from our study, although it may be the case that significant events in the classroom had impinged on a student’s reasoning.17 Most Year 9 students who we interviewed were unable to recall why they had answered as they did in Year 9 let alone in Year 8, and, disconcertingly, when shown their Year 8 responses were often quite ready to switch back to their earlier (incorrect) argument. Clearly this is an area where understanding can be fragile! Perhaps, given that our findings suggest that there is some connection in this particular mathematical context between understanding the structure of implication and successful deductive reasoning, a way forward would be to make the strategies and different profiles of responses explicit in the classroom. The next step would then be to design activities that focus on developing meanings for the structural properties of logical implication in mathematics, and teaching that sustains and develops these meanings over time.

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ACKNOWLEDGEMENTS We gratefully acknowledge the support of the Economic and Social Research Council (ESRC), Project number R000237777. We also wish to thank Einar Jahr of Hedmark University College, Norway, for his detailed comments on the paper from the perspective of a logician.

N OTES 1. Deductive and inductive are being used very broadly here, to distinguish between reasoning based on properties or structural relationships and reasoning based on empirical data. This is not to suggest that these kinds of reasoning are entirely distinct and that, for example, broadly inductive reasoning may not contain deductive sequences, as the analysis by Reid (2002) clearly shows. 2. The questions asked of the students in the two studies were rather different in focus, which may also account for the different result. The focus of the Fischbein work was on whether a particular theorem was true or not. 3. This description is somewhat misleading, as the phenomenon is also widely observed among adults. The reason for this converse error in everyday life might be that conversational discourse often (though not always) carries an unstated second meaning: for example, if we say, “If it is raining, then I carry my umbrella”, then it is reasonable to assume the converse statement as well as the inverse i.e., “If it is not raining, then I do not carry my umbrella”, else you might as well say “I carry my umbrella all the time”. Clearly this is not always the case. O’Brien in fact found that the difficulties he identified could be considerably reduced by putting the implication in a realistic context, such as “If it’s Jim’s car, it is red”, since experience indicates that here the converse statement is clearly not true. 4. Quine makes a similar distinction between what he calls the material conditional and our everyday attitude to the conditional, where, “if . . . the antecedent turns out to be false, our conditional affirmation (of the consequent) is as if it had never been made” (Quine, 1974, p. 19). 5. The particular statements of logical implication used in our study involve quantification, so if one were to view them in terms of formal logical, then strictly speaking this would involve predicate rather than propositional logic. 6. Children’s utterances tend to be cryptic and elliptical, with only as much information revealed as is deemed necessary to be understood at a given moment (Anderson et al., 1997). The same phenomenon was apparent in many of our students’ written explanations, even though the communication here was to a remote audience and noninteractive, where one might have expected students to be more explicit. When coding such explanations it is tempting to read between the lines and to assign underlying strategies to them, a temptation we sought to resist. 7. Multilevel statistical analysis using student data in conjunction with teacher and school data has also been completed for both year groups, though not the focus of this paper. 8. The other combinations may shed light on whether the statement under consideration actually is an implication, but of course only if students can process the combinations effectively.

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9. This test comprised selected items from TIMSS, none of which involved deductive reasoning. 10. A similar question was used in the previous Healy and Hoyles study and was found to have significant outcomes. 11. In our sample, only a handful of students attempted to ‘unpack’ this deduction, by trying to explain why only odd starting numbers produce an odd product or why their sum must be even. 12. In version B of the Year 9 test, L1 appeared at the end of the test and it was noticeable that the number of students who made little or no attempt at this question was considerably higher than in version A (218 compared to 127 students). 1984 students took the Year 8 and Year 9 tests, but to make what we believe to be more accurate comparisons, we present here only the responses of the 1078 students who completed version A of the Year 9 test (and either version of the Year 8 test). 13. Clearly some students were unable to sustain this multiple level of engagement with the question, as revealed by some of the inconsistencies in their responses. We have not attempted to model these inconsistent responses, nor those responses where students seemed confused about some of the basic terms of odd, even, sum and product. 14. In order to select a range of scripts for this further analysis, students’ total scores on the question in Year 8 and Year 9 were used to calculate the percentage increase in the average score for each class; a sample of classes was then chosen which included classes where the percentage increase was very high (greater than 40%) and where it was close to (including just below) zero. All the student responses to L1 as a whole in these classes were looked at again. 15. A very similar pattern of response frequencies is apparent in preliminary analysis of the Year 10 (age 15 years) data. 16. Although it has to be said this might in part be due to an artefact of question L1, namely that the question is framed in terms of starting numbers that affect the sum and product, and hence the antecedent and consequent, ‘simultaneously’. 17. For example, one of our students, when interviewed about his written response, explained that he had used large starting numbers, such as 9 and 17, to test Fred’s statement, “Because I remember doing an experiment before this, which it worked up to a point, then didn’t work over the point. We’ve done one this year as well, it worked up to number seven or something like that”.

R EFERENCES Anderson, R.C., Chinn, C., Chang, J., Waggoner, M. and Yi, H.: 1997, ‘On the logical integrity of children’s arguments’, Cognition and Instruction 15(2), 135–167. Balacheff, N.: 1988, ‘Aspects of proof in pupils’ practice of school mathematics’, in D. Pimm (ed.), Mathematics, Teachers and Children, Hodder and Stoughton, London, pp. 216–235. Bell, A.W.: 1976, ‘A study of pupils’ proof-explanations in mathematical situations’, Educational Studies in Mathematics 7, 23–40. Ceci, Stephen J.: 1990, On Intelligence – More or Less. A Bio-Ecological Treatise of Intellectual Development, Century Psychology Series, Prentice Hall, USA. Cheng, P.W. and Holyoak, K.J.: 1985, ‘Pragmatic reasoning schemas’, Cognitive Psychology 17, 391–416.

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Coe, R. and Ruthven, K.: 1994, ‘Proof practices and constructs of advanced mathematics students’, British Educational Research Journal 20(1), 41–53. Deloustal-Jorrand, V.: 2002, ‘Implication and mathematical reasoning’, Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education 26(2), Norwich, England, pp. 281–288. De Villiers, M.: 1990, ‘The role and function of proof in mathematics’, Pythagoras 24, 17–24. Fischbein, E.: 1982, ‘Intuition and proof’, For the Learning of Mathematics 3(2), 9–18, 24. Godino, J.D. and Recio, A.M.: 1997, ‘Meaning of proofs in mathematics education’, Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, 2, Lahti, Finland, pp. 313–320. Hanna, G.: 1989, ‘Proofs that prove and proofs that explain’, Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education, G.R. Didactique CNRS Paris, pp. 45–51. Hanna, G.: 1995, ‘Challenges to the importance of proof’, For the Learning of Mathematics 15(3), 42–49. Hanna, G.: 2000, ‘Proof, explanation and exploration: an overview’, Educational Studies in Mathematics 44, 5–23. Healy, L. and Hoyles, C.: 2000, ‘A study of proof conceptions in algebra’, Journal for Research in Mathematics Education 31(4), 396–428. Hewitt, D.: 1992, ‘Train spotters’ paradise’, Mathematics Teaching 140, 6–8. Hoyles, C.: 1997, ‘The curricular shaping of students’ approaches to proof’, For the Learning of Mathematics 17(1), 7–16. Inhelder, B. and Piaget, J.: 1958, The Growth of Logical Thinking, Routledge and Kegan Paul, London. Krummheuer, G.: 1995, ‘The ethnology of argumentation’, in P. Cobb and H. Bauersfeld (eds.), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures, Erlbaum, Hillsdale, NJ, pp. 229–269. Mitchell, D.: 1962, An Introduction to Logic, Hutchinson, London. O’Brien, T.C., Shapiro, B.J. and Reali, N.C.: 1971, ‘Logical thinking – language and context’, Educational Studies in Mathematics 4, 201–219. Quine, W.V.: 1974, Methods of Logic, (Third edition; first edition published in 1950) Routledge & Kegan Paul, London. Reid, D.A.: 2002, ‘Conjectures and refutations in Grade 5 mathematics’, Journal for Research in Mathematics Education 33(1), 5–29. Rodd, M.: 2000, ‘On mathematical warrants’, Mathematical Thinking and Learning 3, 22–244. Simon, M.A.: 2000, ‘Reconsidering mathematical validation in the classroom’, Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education 4, Hiroshima, Japan, pp. 161–168. Sowder, L. and Harel, G.: 1998, ‘Types of students’ justifications’, The Mathematics Teacher 91(8), 670–675. Toulmin, S.: 1958, The Uses of Argument, Cambridge University Press, Cambridge, UK. Van Dormolen, J.: 1977, ‘Learning to understand what a proof really means’, Educational Studies in Mathematics 8, 27–34. Vinner, S.: 1983, ‘The notion of proof – some aspects of students’ views at the senior high school level’, Proceedings of the 7th Conference of the International Group for the Psychology of Mathematics Education, pp. 289–294.

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Wason, P.C.: 1960, ‘On the failure to eliminate hypotheses in a conceptual task’, Quarterly Journal of Experimental Psychology 12, 129–140. Wason, P.C. and Shapiro, D.: 1971, ‘Natural and contrived experience in a reasoning task’, Quarterly Journal of Experimental Psychology 23, 63–71. Yackel, E.: 2001, ‘Explanation, justification and argumentation in mathematics classrooms’, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 1, Freudenthal Institute, Utrecht, Holland, pp. 9–23. Yackel, E. and Cobb, P.: 1996, ‘Sociomathematical norms, argumentation, and autonomy in mathematics’, Journal for Research in Mathematics Education 27, 458–477. Zack, V. and Graves, B.: 2001, ‘Making mathematical meaning through dialogue: “Once you think of it, the Z minus three seems pretty weird” ’, Educational Studies in Mathematics 46(1–3), 229–271.

School of Mathematics, Science and Technology, Institute of Education, University of London, Bedford Way, London WC1H 0AL, UK Telephone +44 (0)20 7612 6783, Fax +44 (0)20 7612 6792, E-mail: [email protected]

JANE M. WATSON

INFERENTIAL REASONING AND THE INFLUENCE OF COGNITIVE CONFLICT

ABSTRACT. This study follows two earlier studies of school students’ abilities to draw inferences when comparing two data sets presented in graphical form (Watson and Moritz, 1999; Watson, 2001). Using the same interview protocol with a new sample of 60 students, 20 from each of grades 3, 6 and 9, cognitive conflict was introduced in the form of video clips of reasoning expressed by students in the earlier studies. This methodology was intended to mimic the type of argumentation that might take place in the classroom but in a controlled setting where identical arguments could be presented to different students. Interviews were videotaped and analysed in a similar fashion to the earlier studies in order to document change associated with the presentation of cognitive conflict. Change was documented with respect to the levels of observed response for two parts of the protocol and for the use of displayed variation in the graphs. Implications of the methodology for future research and teaching are discussed.

1. I NTRODUCTION

Following on previous research, this study introduced a new methodology to study students’ development of inferential reasoning. Watson (2001) reported on longitudinal interviews with 42 students after three or four years and found that 62% improved their levels of response. That study provided no instructional intervention over the years and hence natural maturation and engagement with various aspects of the school curriculum are likely to account for the observed change. A question that immediately arises is whether change can be brought about more quickly with some form of educational intervention. This study considered one type of intervention, associated with the presentation of cognitive conflict, as a way of changing conceptions. In the context of asking students to make inferences about two data sets presented in graphical form, cognitive conflict was presented in the form of video clips from the earlier students. Students’ responses before and after experiencing the opinions of other students were monitored to assess the effect of such an intervention. Although not providing a classroom setting within which the interviewees could interact with the other student, this methodology did provide input from students rather than teachers or other adults and it was repeatable over a series of interviews. More control was possible than in a classroom while at the same time still Educational Studies in Mathematics 51: 225–256, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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providing genuine opinions of peers. This study is hence a further step in investigating ways to improve students’ development of understanding of statistical inference. The following two sub-sections highlight previous research associated with students’ understanding of inference and with the use of cognitive conflict to influence students’ choices of more appropriate reasoning.

1.1. Inferential reasoning Making inferences in comparing two data sets is a natural setting within which to present cognitive conflict. A student may or may not initially experience internal conflict in making a decision but once one is made, an alternative can be presented. It is also a relevant topic in the light of current curriculum documents of many countries (e.g., Australian Education Council, 1991, 1994; National Council of Teachers of Mathematics [NCTM], 2000). In the United States, for example, “beginning in grades 3–5 and continuing in the middle grades, the emphasis should shift from analyzing and describing one set of data to comparing two or more sets” (NCTM, 2000, p. 50). Appropriate examples suggested include comparing the sleep habits of first and fifth graders for grades 3–5 (p. 180–181) and comparing flight distances of two types of paper airplanes for grades 6–8 (p. 250). Students’ strategies for comparing two data sets were investigated by Gal and his colleagues (Gal, Rothschild and Wagner, 1989, 1990; Gal and Wagner, 1992). They presented data sets in graphs, containing measurements such as test scores or distances jumped by frogs, to third-, sixthand ninth-grade students in the United States. Three methods for comparison were observed: Statistical strategies, associated with summaries of the data in each group; Proto-statistical strategies, which focused on some but incomplete features, and Other/task-specific strategies, which found totals when they were inappropriate, invented qualitative or idiosyncratic explanations. Estepa, Batanero and Sanchez (1999) asked senior secondary students to make comparisons of two small data sets (n = 10 each) with numeric values presented in tables. Four correct strategies observed were: comparing means, comparing totals, comparing percentages and comparing the two distributions. Four partially-correct strategies were comparing each pair of values in related samples, taking into account exceptional cases, finding out the differences for all pair-wise cases and carrying out vague global comparisons. Five strategies considered incorrect were using a deterministic stance to expect similar values in the two data sets, comparing highest and lowest values, comparing ranges, assessing coincidences

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in the data, and basing conclusions on previously held beliefs about the context of the data. Konold, Pollatsek, Well and Gagnon (1997) and Biehler (1997) were also interested in the reasoning senior secondary students used in making decisions about data sets with the option of using the data handling software DataScope (Konold and Miller, 1994). Among the many concerns expressed about students’ performances, the major one was related to the tendency to focus on individual features rather than global properties in decisions. This was illustrated for example in choices to consider data in frequency tables and focus on individual cells rather than to represent data in graphs to gain information from the complete distribution of values. Further work of Konold, Robinson, Khalil, Pollatsek, Well, Wing, and Mayr (2002) with middle school students has focused on a tendency to interpret stacked dot plots based on clumps of data values observed in the plot rather than formal measures of centre. The observations of Konold et al. (2002) highlight one aspect of the current surge of interest in school students’ understanding of variation. In response to the suggestions of Green (1993) and Shaughnessy (1997) that variation become a focus of research, recent studies have considered variation in sampling situations for chance settings (Shaughnessy, Watson, Moritz and Reading, 1999; Reading and Shaughnessy, 2000; Torok and Watson, 2000; and Kelly and Watson, 2002) and specifically in relation to decision making in comparing two groups (Watson, 2001). This last research, the precursor to the current study, reinforced not only the concerns of Konold et al. (1997) and Biehler (1997) with respect to many students’ focus on individual features of the graphs but also the observations of Konold et al. (2002) with respect to global features that in many cases were similar to clumping. The range of outcomes observed by Watson engendered continued interest in the effect that cognitive conflict might have on students’ use of variation in their decision making in comparing two groups. 1.2. Cognitive conflict In learning situations, cognitive conflict can take many forms; it may, for example, arise naturally when one’s guess or hypothesis is proved wrong after the outcome of an experiment or it may be provoked by a teacher or another student stating a contrary opinion. For learning to occur, however, the conflict must create dissatisfaction with the original belief and the alternative view must be intelligible and plausible (Strike and Posner, 1992). Science educators believe that in experiencing such conflict student learners are in a similar situation to the wider science community

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as it advances (Posner, Strike, Hewson and Gertzog, 1982). In fact it is from science education research that many of the suggestions for providing teaching scenarios to promote cognitive change come (Driver, 1985; Macbeth, 2000). Although based to a great extent on presenting empirical counter examples, along with opportunities to review alternative conceptions, science educators also stress the necessity for students to make their own conceptions explicit so that they are available for change. Mathematics educators have considered similar issues in attempting to help students construct appropriate mathematical understanding (e.g., Tirosh, Graeber and Wilson, 1990; Steffe, 1990). In addressing the issues of establishing an appropriate level of cognitive conflict and guiding students to conflict resolution, Behr and Harel (1990) suggest devising specific experiences based on children’s knowledge structures, on appropriate mathematical domains and on the cognitive structures necessary for these domains. Shaughnessy’s (1985) suggestion for emphasising conflict by getting students to express clearly their own ideas first and then be exposed to situations where they fail is consistent with the above frameworks. In the area of probability for example students have been asked for a prediction for a chance outcome and then given the opportunity to test the prediction through an experiment (Pratt, 1998; Shaughnessy and Ciancetta, 2002; Kelly and Watson, 2002). Group work, as well is often suggested as a means of encouraging sharing of ideas and potential cognitive conflict among peers. To ensure conflicting ideas lead to appropriate learning, theorists such as Vygotsky suggest organising groups in such a way that the range of intelligible ideas includes those at a slightly higher level than those of the targeted students (the Zone of Proximal Development) (Goos, 2000). In an actual classroom setting, however, there is no guarantee that the appropriate conflict will be provided or that the targeted students will initially express their beliefs clearly. In a research interview setting there is a greater chance of having the interviewees state their initial understanding clearly and then of choosing an appropriate form of conflicting opinion. In the research setting there is the further possibility of challenging the higher-level ideas of students with less suitable alternatives in order to test the stability of the ideas for these students. Defending one’s beliefs, when appropriate, is another benefit of being placed in a situation of cognitive conflict. Other research as part of the larger project which included this study, used this methodology with the same students to explore understanding of chance measurement (Watson and Moritz, 2001a), sampling (Watson, 2002) and pictographs (Watson and Moritz, 2001b). For chance measurement approximately one-third of students who could improve their per-

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Figure 1. Graphs used for the four parts of the interview protocol.

formance did so after hearing conflicting opinions of other students. For sampling the improvement rate dropped to 22 percent. Conflict was introduced for two aspects of reasoning with pictographs, with 60 percent of grade 3 students preferring higher level representations suggested by other students but only 30 percent preferring higher level predictions. These outcomes encourage interest in using the methodology on other topics, to gather more data on improvement rates and to hypothesise reasons for the range of improvement rates.

2. BACKGROUND FOR CURRENT STUDY

Watson and Moritz (1999) presented 88 third- to ninth-grade students with graphs of four pairs of data sets shown in Figure 1 involving quick recall of mathematics facts of children in fictitious classes. The four parts, presented on different pages, each asked for a comparison of two data sets presented graphically and representing classes’ scores on a mathematics

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quiz. For Part (a) the interviewer read along with the student the following information. Two schools are comparing some classes to see which is better at quick recall of 9 maths facts. In each part of this question you will be asked to compare different classes. First consider two classes, the Blue class and the Red class. The scores for the two classes are shown on the two charts below. Each box is one person’s test, and the number inside is their score. In the Blue class, 4 people scored 2 correct and 2 people scored 3. In the Red class, 3 people scored 6 correct and 3 people scored 7.

Students were then given the task instructions: Now look at the scores of all students in each class and then decide, did the two classes score equally well, or did one of the classes score better? Explain how you decided.

A neo-Piagetian model of cognitive functioning (Biggs and Collis, 1982; Biggs, 1992) was used as a theoretical framework for describing three response levels. Unistructural responses (U) involve only one relevant aspect of the task presented. Multistructural responses (M) involve several disjoint relevant aspects, usually in sequence, but not all aspects are integrated. Relational responses (R) demonstrate an integrated understanding of the relationships among all aspects of the task. For tasks of comparing two data sets, Watson and Moritz (1999) assessed students’ reasoning in two such U-M-R cycles. In the first cycle, responses compared data sets of equal sizes, with or without success, but they did not recognize and/or did not resolve the issue of unequal sample size. U1 M1

R1

A single feature of the graph was used in simple group comparisons; e.g., “Red got more points.” Multiple-step numerical calculations or visual comparisons were performed in sequence on absolute values for simple group comparisons, not necessarily resulting in the correct conclusion; e.g., calculating totals for each group, or looking in turn at different values of the data sets and comparing frequencies. All available information was integrated for comparisons of groups of equal size, including both visual observations and numerical justification; appropriate conclusions were restricted to comparisons with groups of equal size; e.g., visually identifying values for which the groups had different frequencies and then totalling the values of these differences to assess which had the highest total of all scores.

In the second cycle, the issue of unequal sample size was resolved with some proportional strategy employed for handling different sizes.

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M2

R2

231

A single visual comparison was used appropriately in comparing groups of unequal sample size; e.g., “Black have got a higher amount for the number of people.” Multiple-step numerical calculations or visual comparisons were performed in sequence on a proportional basis to compare groups; e.g., calculating two means and comparing them. All available information, from both visual comparison and calculation of means, was integrated to support a response in comparing groups of unequal sample size.

For third-grade students, the highest level of response was M1 . For fifthto seventh-grade students, responses at all levels were observed with most occurring at M1 or R1 . For ninth-grade students, responses were distributed across the M1 to R2 levels with more higher-level responses than for younger students. The framework employed in the original study and the range of responses observed appear to satisfy the preliminary suggestions of Behr and Harel (1990) for establishing and resolving cognitive conflict with respect to inferential reasoning. Some students did not recognize that in the last pair of data sets, the two classes were not of the same size. Many of these students were prompted by the interviewer with this information, in a precursor to formal presentation of cognitive conflict, and asked whether they wished to reconsider their responses. Responses to the probing question were classified and about one-third of students responded to the prompt at a higher level than their earlier response, one-third, at the same level and one-third, at a lower level. Also documented by Watson and Moritz (1999) were numerical and visual strategies, which occurred at various levels of reasoning. At the multistructural response levels, 47 percent of responses were based on visual strategies using graphical information and 53 percent on numerical strategies such as totals or averages (n = 58). Overall 18 students responded at the two Relational levels (20% of all students) and these students employed both visual and numerical strategies. In the longitudinal follow-up study of Watson and Moritz (1999), Watson (2001) reported on the responses of 42 students re-interviewed threeor four-years subsequent to initial interviewing. Many students improved their level of reasoning, providing further evidence that the response levels identified by Watson and Moritz constitute a useful model to describe cognitive development on these tasks over the school years. Only four students responded at a lower level, in each case changing from the R level to the M level in one of the two cycles; these changes were explained as due to students’ lack of motivation to provide an optimal response involving all aspects of their reasoning when a simpler response was adequate. Further

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in considering the note made of variation in students’ arguments about which class had done better in Parts (c) and (d), Watson found that 64 percent used the variation in the graphs in Part (c) and 88 percent, in Part (d), in their initial interviews. In the longitudinal interviews 69 percent in Part (c) and 95 percent in Part (d) mentioned variation in graphs. There was a tendency for students to use more sophisticated arguments based on variation after three or four years but 62 percent for Part (c) and 48 percent for Part (d) responded at similar levels each time.

3. R ESEARCH QUESTIONS

Given the previous research interest in the role of cognitive conflict in developing student understanding and the previous research on students’ inferential reasoning using the protocol in Figure 1, it was possible to state research questions reflecting continued interest in students’ initial understandings, in their reaction to cognitive conflict for Parts (c) and (d) of the protocol and in their change in strategy in taking into account the variation displayed in the graphs after being presented with conflict. The research questions for this study centre around the following issues in relation to previous research and the provision of cognitive conflict. 1. What is the distribution of the initial levels of response for the students in the current study compared with that observed by Watson and Moritz (1999)? 2. For responses to Part (c) of the protocol, of the students who responded “Yellow” or “Brown”, how many changed to “Equal” after the presentation of a conflicting view? Of those who said “Equal” and were prompted with a conflicting response, how many changed? 3. For responses to Part (d) of the protocol, of the students who responded “Pink” or “Equal”, how many changed to “Black” after the presentation of a conflicting view? Of those who said “Black” and were prompted with a lower level response, how many changed? 4. Do the responses of students with respect to the variation displayed in the data sets fit within the developmental framework related to individual and global features described by Watson (2001)? What changes occur, after the presentation of cognitive conflict, in student responses related to variation?

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Figure 2. Prompts displayed to students to promote cognitive conflict for Part (c).

4. M ETHOD

4.1. Sample Individual interviews were conducted with 60 students from four Tasmanian government schools: 10 third- and sixth-grade students from each of two primary schools and 10 ninth-grade students from each of two secondary schools. Students were selected on the basis that they would be willing to talk in interview and not be threatened by the complexity of listening to other students’ ideas and evaluating them. Hence it might be expected that the students interviewed were more willing and able than would normally be expected for their grade levels. All interviews were video taped. 4.2. Procedure Following the data collection of Watson and Moritz (1999), a digitised video clip research resource was created using selected student responses from the 88 student interviews. The objective was to present new students being interviewed with conflicting ideas selected from interviews with the earlier 88, mimicking to some extent an ideal classroom setting where students engage in dialogue and debate. Eight prompts were included in the final protocol.

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Figure 3. Prompts displayed to students to promote cognitive conflict for Part (d).

Interviews were during class-time (40 mins.) and the protocol in Figure 1 was the second of five or six protocols involving cognitive conflict; other topics were comparing chances, average (presented after the protocol in Figure 1), sampling, and dice outcomes. The first two parts presented comparisons that were easy to make visually. In this study the first was always used as an introduction to the protocol but the second was usually omitted as repetitive. At two points in the protocol (after Parts (c) and (d)), students were shown digitised extracts (5–30 seconds duration), on a laptop computer, of other students answering the question in a different fashion than the interviewees had done. The extracts used for Part (c) are shown in Figure 2 and those for Part (d) in Figure 3. Interviewees were then asked for their responses to what the other students had said. Often several alternatives were offered to the interviewee. The students were then asked for a final decision for which class did better in each part. Each prompt also included the words said by the interviewer, “What do you think of his/her idea?” Prompts were shown at the interviewer’s discretion, depending on the student’s initial response, in an attempt to create cognitive conflict. All students who initially responded with an incorrect conclusion were prompted with at least one other response, including the correct one. Some students who initially responded correctly were not prompted further for Part (c). For Part (d) this happened for only one person.

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4.3. Analysis The digitised interview videos were transcribed and linked via a spreadsheet both to the transcripts and to the other video clips that were shown to the students during the interviews. Hence the analysis could be carried out with instantaneous access to written and video material of the interviewees as well as to the archived clips. The spreadsheet also contained cells noting which prompts had been used for each student, any change in strategy use, whether the student referred to “total,” “average,” or “percent,” and whether the student discussed the shape of the distributions before and after prompting. All students’ responses were coded in four stages reflecting the research questions: the initial overall SOLO levels of response based on the framework of Watson and Moritz (1999); their specific responses to Part (c) before and after prompting (“Yellow,” “Brown,” or “Equal”); specific responses to Part (d) before and after prompting (“Pink,” “Equal,” or “Black”); and specific mention of variation throughout Parts (c) and (d). A descriptive clustering procedure (Miles and Huberman, 1994), similar to that described by Watson and Moritz (2001) was employed by the author to determine codes based on initial assignments by a second researcher for the first three research questions. For the fourth question the procedure followed by Watson (2001), based on the framework of that study, was employed to categorise the responses by the features of variation considered in making initial decisions for Parts (c) and (d) of the protocol and in defending final decisions involving variation after the presentation of cognitive conflict. For clarification in relation to the coding related to Part (c) where the two classes had the same total and centres but different spreads, it was possible for a student to consider all three possibilities for the classes and not wish to make a final decision. These responses were placed in the “Equal” category as representing statistically appropriate considerations. Responses choosing “Yellow” or “Brown” without equivocation, although not “wrong,” were considered less appropriate in not also considering statistically appropriate aspects of the representations. Tables will present descriptive data in relation to all Research Questions. For Research Questions 2 to 4, examples of responses will be given to illustrate the outcomes. 5. R ESULTS

5.1. Research question 1: Distribution of initial responses The distribution of the 60 initial responses by grade for the overall SOLO level of response using the criteria of Watson and Moritz (1999) is given

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TABLE I Initial responses by grade and SOLO level Level

U1

M1

R1

U2

M2

Grade 3 (n = 20) Grade 6 (n = 20) Grade 9 (n = 20)

2 2 0

16 4 9

2 8 7

0 2 3

0 4 1

in Table I. In comparison with the earlier study, for grade 3 students the distribution of responses was similar with all responses being classified in the first U-M-R cycle. The grade 6 students performed slightly better than the middle school group (grades 5, 6 and 7) earlier, with a larger proportion of R1 level responses and 30 percent in the second cycle, compared to 17 percent. The grade 9 students, however, performed less well with an 80–20 split between the first and second cycles, compared to a 50–50 split earlier. A comparison of the distributions of levels for all students for the two studies (n = 88 and n = 60), with the M2 and R2 levels combined, indicated similar proportions at the five levels (χ 4 2 = 5.21, p > .25). No analysis of responses after interviewer probes was conducted in this study (as was done in Watson and Moritz) because of the cognitive conflict presented by other students. 5.2. Research question 2: Cognitive conflict for Part (c) The initial responses for Part (c) were categorised as “Yellow,” “Brown,” or “Equal,” associated with whether one class was judged as having done better or whether the classes had done equally well. As can be seen in Figure 1, the scores for each class summed to 45 but there was a difference in the shape of the distribution – the Brown class had scores of 3 and 7, with three 5s, whereas the Yellow class had five 5s and no 3s or 7s. Of interest were the arguments used to justify a decision on which class had done better and if the initial response was not “Equal,” whether the student accepted arguments for this decision. Not all students who responded “Equal” initially were prompted with the other responses shown in Figure 2. All students (23) who responded “Yellow” or “Brown” were prompted with at least one other response for the other class. No students changed their decision from one class to the other; they either kept their original response or changed to “Equal.” Table II contains the information at each grade level for each initial response, the number of these who received a prompt from another student

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TABLE II Initial responses to Part (c) and final responses after conflicting prompt, by grade

Grade 3 Grade 6 Grade 9

Initial Response “Equal” n Prompt Final=

Initial Response “Yellow” n Prompt Final=

Initial Response “Brown” n Prompt Final=

12∗ 14 11

5 5 6

3 1 3

3 4 5

3 4 5

5 5 6

3 2 4∗

3 1 3

1 1 2∗

∗ In each cell there was one response showing indecision that was placed in this category

after careful consideration of the video extract.

in Figure 2 and the number of these who were persuaded to change their response to “Equal.” Overall Stan’s “Brown” prompt was given 22 times; Stacey’s “Yellow” prompt, 7 times; Sally’s “Equal” prompt, 20 times; and Stephanie’s “Equal” prompt, 6 times. As can be seen in Table II, over half of students at each grade level initially decided that the classes had done equally well. Of these only 12 were prompted with “Yellow” or “Brown” extracts and none of these changed their views. About a quarter of students at each grade level initially chose the “Yellow” class as having done better. Of these students, just over half were convinced by an “Equal” argument. Only seven students overall initially decided that the “Brown” class had done better and of these, four changed to an “Equal” view after prompting. Hence 57 percent of students who could improve their response level after the presentation of cognitive conflict did so. Several extracts will be presented to illustrate the interaction of the interviewees with the conflict. A grade 3 student who was not persuaded to change from a choice of the “Yellow” class gave the following initial argument and reactions to two other students. S1:

Yellow . . . because they got more 5s [Yellow] even though this person got a 7 [Brown], this score’s better [Yellow] because they got five 5s. Stan’s prompt. I still think that 5 is worth more than 7 because five people did really good and only three people, well three people got, this class [Brown] was very close behind. Sally’s prompt. Pretty good [nods head]. I: So do you think they did the same or Yellow did better? Well the Yellow doesn’t have a 3 which means it probably makes up for the 7 because this class [Yellow] didn’t get a 3 but this class [Brown] got a 7 and this class [Yellow] got more 5s so it has done better.

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On the other hand, “Brown” was the consistent response of the following grade 9 student. S2:

The Brown class . . . Because, although more people here [Yellow 5] got 5 here’s [Brown] a more even balance. Sally’s prompt. Yes pretty good. I: So do you think they would have done equally well or did Brown do better? Brown probably did a bit better but if you just take it, then Brown. I: So you don’t go for her argument . . . that sort of evens it out? Well they might have been about the same but this looks like the class [Brown] is a bit more even than if there were good people in there and worse people.

Some students who swapped to “Equal” from another view offered little in the way of further justification from that given by the other student, as was the case for the following grade 6. S3:

Yellow. I: Why? I think there is more there. More 5s. Stan’s prompt. No, I would go with Yellow. Sally’s prompt. Yes, that’s alright . . . I think they scored equally well.

Other students, however, restated arguments and clarified their positions. S4:

[Pause] I would have to say Brown. I: Why? Because they [Brown] not only got higher results than the others, they got more varied and spread out. Sally’s prompt. Yes it is quite right because you have got the 7 and the 3 which pretty much equals up to the two 5s in the Yellow. So yes, that would be also kind of correct. I: So is she right or would Brown have done better? I reckon she is right in that they scored equally and also right that Yellow or, whichever one I said got better, did better because, yes. I: First you said Brown . . . would you still say Brown or did they do equally? Probably say change to that they are pretty equal. Yes. [grade 9]

5.3. Research question 3: Cognitive conflict for Part (d) The responses to Part (d) were categorised as “Pink,” “Black,” or “Equal,” depending on whether one class was chosen as having done better or whether the class did equally well. As can be seen in Figure 1, the “Pink” class had more students but the distribution of scores in the graph indicates that the “Black” class produced better scores on average. There were two prompts favouring the “Pink” class and two favouring the “Black” (Figure 3). Samuel’s prompt was given 7 times; Samantha’s, 30 times; Simon’s, 48 times; and Selina’s, 23 times. For this part all students were prompted except one in grade 6 who initially responded “Black” and compared the arithmetic means for the

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TABLE III Initial responses to Part (d) and final responses after conflicting prompts, by grade Initial Response Initial Response Initial Response “Pink” “Equal” “Black” Final Response Final Response Final Response n “Equal” “Black” n “Pink” “Black” n “Equal” “Pink” Grade 3 181 Grade 6 11 Grade 9 14

3 0 0

3 4 6

22 33 24

0 0 0

0 2 0

0 65 4

0 0 0

0 0 0

1 One student initially wavered “Pink,” “Black,” then “Pink.” 2 One student initially wavered “Equal,” “Pink,” “Equal.” 3 One student hovered between “Pink” and “Equal,” and after prompts, between “Pink”

and “Black.” 4 One student initially wavered “Equal,” “Black,” “Equal.” 5 One student not prompted.

two groups. Table III hence summarises the initial response and final response after prompting if it represents a change. In grade 3 for example, 18 students initially said the “Pink” class did better, with 3 of these changing to “Equal” and 3 to “Black” after the prompts were given; thus 12 grade 3 students were not influenced by the prompts to change their responses. Overall 72 percent of students initially chose the “Pink” class, with 12 percent saying the classes had done equally well, and 17 percent choosing the “Black” class. Following the presentation of cognitive conflict to the 50 students who did not say “Black”, 30 percent of these students agreed with arguments that the “Black” class had done better. It is of considerable interest then to examine the transcripts to understand why so many students did not change from their initial views and what may have influenced some to do so. Many of the students who chose the “Pink” class noticed there were “more” in that class, either more Xs or a higher total score and were not influenced by the conflicting arguments of other students. The following grade 3 student is typical.

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I reckon that class [Pink]. I: Why? Because they’ve got more things [Pink columns 5 and 6 correct on sheet] and that’s a better class. Simon’s prompt. No because I reckon he’s wrong because they’ve [Pink] still got higher. They’ve done higher than them [Black] because they [Black] haven’t got one up the top yet and we’ve got two [this is referring to the centre top of the distribution not to high scores]. I: So you’d say Pink? Yes.

Some arguments were more complex, employing different types of reasoning in the face of different class sizes. S6:

S7:

Definitely the Pink because they [Pink/Black] have got even until them [Pink middle] which is 42 and 35, which is actually 77, which they [Black] don’t really, they can’t . . . they can probably only just make 7’s. 77 is just that [Black right] and so they [Pink] have still got that to work with [Pink left]. So easily the Pink, could get that. I: Does it matter Pink has more students? Well, it would kind of affect it because the Pink would have an advantage then because like if they [Black] had more people, more people could have got 9 or 8, or 7 to make the score go up. So Pink does have a bit of an advantage. So it does make a bit of a difference, yes. I: Would you still say Pink or . . . ? If it [Black] had more people it would probably be more even. But Pink definitely has a better score than Black because of a lesser amount of people. Simon’s prompt. Yes because they have done reasonably well to get six people on 7 and four on 6 and 8, they have done reasonably well to get that. So they could have all got 1, 2, or 3, but they have done reasonably well to get that. But I still reckon Pink. If Black had more people I reckon Pink would still have had a better score. [grade 6] Pink . . . because they have higher ratings around the 5 and 6 whereas these guys [Black] only had 2s and 4s and they [Pink] have 7s there. So definitely Pink because of the lower values around here [Black left] and Pink is really up. It is definitely Pink. That is another obvious one. Simon’s prompt. I thought what you were saying since there is less people in the class they would be better . . . for how many people there are in the class. I: Yes for how many people. But that’s not the question. The question is who did better in all through so. I: So you would still say Pink did better? Yes. I: Do you think it does make a difference when they do have different numbers of people? Yes it would make a difference with the number of people because more people you can get more marks but that still doesn’t change the fact that Pink did better in the scores, you can’t say they didn’t. Selina’s prompt. Well she basically said the same thing as the other guy did. I: Still Pink? Yes. [grade 9]

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TABLE IV Strategies used in comparing data sets Numerical Total Average Grade 3 Grade 6 Grade 9

9 6 4

0 12 7

Visual/ shape 14 18 19

The response saying the two classes were “Equal” occurred more often in this study than in the earlier ones. The following grade 9 student’s response is typical of these.

S8:

Is there more people in that one [Pink]? I: Yes. There’s 38 people there and just 21 in the Black class. Probably about the same. I: How are you deciding that they are about the same in this case? Well this [Black] has got less people but they [Pink/Black right] are still the same. Like most of them have got kind of the same scores, this [Black] has just got a few lower I reckon. Samantha’s prompt. I guess it depends if you are saying about majority or if you are just saying how people scored individually comparing them. I: And so if you went by majority you would like that argument? Yes. I: But if you went by the scores you would say they were about equal or . . . ? Yes. Simon’s prompt. It is a good point because they have more people in the 6, 7 and 8, and the Pink class has more people down there [Pink left]. But I mean are you going to have more people down there if you have more people anyway. I: Sorry? If you have got more people then you are going to have a bigger variety anyway? I: So do you think his case is very good, I mean you said that some bits of it you liked. [nods head] I: Do you think that is a good argument to say that Black scored better? Yes there’s good arguments for both of them though. So I guess I have to say equal because I am not really sure. [grade 9]

Those who accepted the views of the other student that the “Black” class had done better sometimes were sensitive to the class size issue when it was raised and could stay with the better view when given another “Pink” argument.

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S10:

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Probably the Pink class. I: Why? Because they got more people in like the 4 and the number correct 4 and 3 and 5 and so more people got those correct so that would make them more. Simon’s prompt. Yes he has actually got a fairly good point . . . yes considering on how many people there is. Yes. I: Do you think it matters that the Pink does have more people? Umm. I: Does that influence your decision on which one did better? Well in some cases yes but like he said there’s less people in the Black class so they have performed better in a way. So yes. I: So has he made you change your mind to Black? Yes to the Black. Samantha’s prompt. No I actually think Black. Yes, still think Black now. I: So the other one was more persuasive then? Yes. I: You changed your mind initially. Can you identify what it was that made you change your mind? The fact that Black has smaller people, like smaller amount of people, so therefore they have performed better to get like that amount of number correct for the number of people they got. Whereas Pink has got heaps of people so their percentage would be bigger anyway because they have got more people and I don’t know. I: So the percentage would be bigger or something? How would you measure what that score was? Umm . . . probably by the number of people that went in it I guess. I don’t know. [shrugs shoulders] Something like that. Yes the people. [grade 9] I would say Pink. I: Why? Because it just looks like there’s more. Simon’s prompt. I think it is probably better than mine. I: So how is he arguing there with the argument? He is arguing that the average of the Black is better than the Pink. I: So what do you think the average would be for those? Probably about 7 for the Black and 5 or 6 for Pink . . . I: What made you change your mind there? I mean you heard what he said but . . . Well, umm . . . probably Pink is most likely to have more because there’s a lot more people but because there’s less Black it looks like there’s less but they did score a better average. Samantha’s prompt. I still reckon Black. [grade 9]

5.4. Research question 4: Variation before and after cognitive conflict As a preliminary analysis for the consideration of students’ use of variation in their arguments, it is necessary to describe their overall use of numerical and visual strategies in the comparisons. Watson and Moritz (1999) reported on the use of numerical and visual strategies for students who responded at the multistructural levels and Watson (2001) reported on students employing numerical, visual and mixed strategies in their decisionmaking. In the current study three students did not display overtly either numerical or visual strategies, whereas 20 used both. A few also used both totals and averages in their responses. Particularly for the numerical strategies these were employed principally in the initial part of the

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TABLE V Strategies for using variation in comparing data sets (n1 , n2 ) = (initial, after conflict) No acknowledgement of variation Part (c) Part (d) (14, 13) (12, 20) Individual features

Global features

Single column(s) considered: less than or equal to 2, or no synthesis

“More” with no justification

Part (c) (6, 1)

Part (c) (5, 3)

Part (d) (1, 1)

Part (d) (7, 1)

More than 2 columns considered (but only columns)

Multiple features considered: global or global plus columns, sequential analysis

Part (c) (18, 9)

Part (c) (12, 4)

Part (d) (13, 4)

Part (d) (20, 21)

Multiple features considered: integrated, compared and contrasted Part (c) Part (d) (5, 5) (7, 12)

interview before the introduction of other students’ views. Hence the data reported in Table IV reflect all comments during the interviews, not differentiated as before or after the experience of cognitive conflict. Table IV reports on use of numerical strategies, including average and total, or visual strategies related in some way to the shape of the distributions. The word “average” is used in this context rather than “mean” because students rarely used the word “mean” and in using “average” they often used it in a general sense with no explanation or with an informal middle or modal treatment. In the initial part of the interviews 47 students mentioned visual aspects (11 grade 3, 17 grade 6 and 19 grade 9). The use of visual strategies was common and increased slightly over the grades, with all but one grade 9 student making some visual observation. In the clustering procedure that was used in the analysis of student responses with respect to variation, Parts (c) and (d) of the protocol were considered separately. The categories used were those of Watson (2001) and all responses could be accommodated under the scheme. The six categories reflected attention to three aspects of features of variation noted in

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the graphs: no features, individual features and global features. These six categories are laid out in Table V in a hierarchical fashion reflecting the complexity of the arguments made. The first category contains responses that do not acknowledge that variation is shown in the graphs. For the two individual features categories, responses in the lower grouping either mention one or two columns only or show no synthesis in describing columns. In the higher grouping, more than two columns are mentioned sequentially. For the three global features categories, three levels of grouping emerge. The lowest level responses mention “more” in reference to the Pink class with no further justification. At the next level, multiple global features (and perhaps including columns) are considered in a sequential fashion. At the highest level multiple features are integrated, with consideration including comparisons and contrasts. As can be seen in the table, the initial two levels of response for individual and global features are considered to be structurally equivalent. For each of Parts (c) and (d) in Table V, the first number represents the number responding initially in that category; hence the first numbers sum to 60. For each part, the second number represents the number responding in that fashion after being introduced to cognitive conflict; hence the second numbers sum to 35 for Part (c) and to 59 for Part (d). For the global features for Part (d), the highest category (multiple features considered: integrated, compared and contrasted), the numbers of responses represent those whose argument included proportional reasoning based on the shape of the graphs and concluded that the Black class had done better. Of the students who did not acknowledge variation in Part (d) (n1 = 12, n2 = 20), three in the initial part of the interview and eight after the prompts of other students also agreed that the Black class had done better. These students either suggested or accepted using the arithmetic mean with no reference to the graphs or agreed (e.g., “I like his argument”) with no further reasoning. Table VI summarises for students who were shown conflicting views of other students, the association of their acknowledgement of variation before and after the prompt. As can be seen in the table it appears that quite a few students who used arguments including variation before seeing a conflicting view, did not feel the need to repeat or expand on that argument after the prompt: 11/29 for Part (c) and 13/48 for Part (d). On the other hand 9 students for Part (c) and 18 for Part (d) did present global arguments each time. Of interest in the light of responses reported in Watson (2001) illustrating the five categories that took note of variation (Table V), are the reactions of students when prompted with cognitive conflict. Some examples associated with the categories will be presented, including others quoted earlier.

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TABLE VI Association use of variation of initial and after-conflict responses for Parts (c) (n = 35) and (d) (n = 59) of the protocol Part (c)

Initial comments on variation

After conflict No variation Individual features Global features

No variation 2 3 1

Part (d)

Initial comments on variation

After conflict

No variation

“More”

7 0 2 2

4 0 1 3

No variation “More” Individual features Global features

Individual 8 3 2

features

Global features 3 4 9

Individual features 3 1 0 10

Global features 6 0 3 18

5.4.1. Individual features – single columns Only one student focused on a single column both before and after prompts from other students. S11:

I reckon Brown did a bit better because one person got the higher [Brown 7], higher than two people [Yellow 6]. Sally’s prompt. Not really. Stacey’s prompt. I: So do you think the Brown did better or did they do equally or . . .? I think the Brown did better still because they [Brown] got higher scores than the Yellow. Stephanie’s prompt. I didn’t realise that but that’s probably a good idea because they have both got exactly the number of scores but I still reckon Brown is better because they got a higher score [Brown 7]. [grade 3]

Student S3 was one who originally used a single column argument but did not refer to variation after a prompt for Part (c) and S5 focused on single columns in Part (d) before switching to a “More” argument. 5.4.2. Individual features – multiple columns Although many students mentioned multiple columns in their initial responses, many were not shown conflict or did not feel the need to repeat their arguments. One student consistently referred to multiple columns in her responses.

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I: And why do you say Yellow? Because Brown hasn’t got all the 5s up here and you can see here at the end [Brown 3] they haven’t got it [Yellow] so I think it would be that [Yellow column of 5s] because if you put those two together [Brown 3 and 7] it would make the same amount [Yellow 5s] like going up, it would be same height. Sally’s prompt. I think hers is a little better. . . . Yes because like it’s really, because they are equal because they’ve [Yellow] got a lot of 5s and 6 and 4s, but that one [Brown] has got a 7 and a 3 and that one hasn’t, so it would be a bit in between. [grade 3]

5.4.3. Global features – “More” Although seven students used a base “More” argument initially for Part (d), none did after a conflicting prompt. One of these was S10, quoted earlier as saying, “Pink, because it just looks like there’s more.” As noted S5 was the one student who used this argument after prompting. 5.4.4. Global features – multiple features Several students consistently noted multiple features related to variation before and after conflicting prompts for Part (c).

S13:

Yellow class I would say because it has a more even group [indicates with hands] unlike the Brown class which has a very intelligent student [Brown 7] and a not so intelligent student [Brown 3] who gets only 3. I would say the Yellow class is much better educated because there is an average there, usually the average is 5. So I would say they did. I: So they [Yellow] have an average of 5? Yes. I: Could you describe them [Brown] having an average as well? They could have an average of 5, yes, they do. But I still say it [Brown] is a bit more uneven that class, unlike the Yellow. . . . Stephanie’s prompt. A very logical conclusion. I would agree with her. And umm if . . . yes I would agree with her but I would still stick with Yellow, that the Yellow class did better because it is an average class solution. I mean they [Brown] are a bit more random here whereas they [Yellow] are a bit more straight forward there, fairly equal. [grade 9]

Student S8 was one who used similar multiple features argument twice in Part (d) to justify responses of “Equal.” 5.4.5. Global features – integrated, compared and contrasted One student consistently used arguments that compared and contrasted the sets in Part (c).

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[Add both groups to get 45.] Can you say equal? I: You can say they are equal if you think they are equal. Do you think they are equal? You got the same result. Umm . . . actually I reckon that Yellow because their class had a better average altogether than like they [Brown] have got a 7 which is, they [Yellow] don’t have a 7 but they [Brown] have got like 3 and they [Yellow] are more consistent around [Yellow 5], more consistent I reckon. So I reckon that Yellow is better. I: Which scored better, the Brown class or the Yellow class? I wasn’t sure about equally well but I thought on an average that the Yellow would be better because they keep a consistent score [Yellow] and that they [Brown] have got like 3, 4, 5, 6, 7 . . . 4, 5, 6 [Yellow] so they get a consistent. Stan’s prompt. It is only like one 7 [Brown] but if you plus 7 and 3, they only make two 5’s [Yellow], so that’s still average are the same. The 7 doesn’t really mean anything if you have got a low number. So they could be like the same or I thought Yellow because they are more consistent with their scores. [grade 6]

For Part (d) some of the arguments that considered the proportional features of the graphs were quite succinct, both before and after prompts. S15:

S16:

I would say this one did better, the bottom one [Black]. I: Why do you say that? Well it is like more forward [Black right columns], like the big point in it [tallest Black and Pink] – the one with the most. Samuel’s prompt. I reckon that’s the bottom half of the graph though. So I would say the average is about here [Pink 5] and this half [Pink left] has done lower than kind of what that half has [Pink right] and this [Black right] has got more in that half. [grade 6] Well I think it would be hard to compare because it seems these [Pink] have got more people. So it is going to be harder to compare than average scores. . . . I reckon these ones [Black] because the high scores are like more marks, because like this one [Pink] high scores are in the middle. Samantha’s prompt. Well, she is right that they [Pink] scored well there but the majority is like the people . . . if you compare them so they have the same amount of people, like double it or something these [Black] would end up with the best score I reckon because they have got more people with high scores. [grade 9]

6. D ISCUSSION

The discussion considers the research questions in turn and then the use of the methodology for studying the development of students’ inferential reasoning. Implications for teaching and future research are also included.

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6.1. Research question 1 – Distribution of initial responses Comparing the distribution of levels of the 60 responses in the current study with the 88 in Watson and Moritz (1999) indicated a similar representation across the development levels, except that no R2 responses were observed in this study. The lack of R2 responses (5% in the earlier study), those including both a numerical (e.g., mean) comparison and a visual comparison (e.g., placement of scores) for Part (d), may have resulted from a perception on the part of the students and the interviewer that providing a single justification was sufficient in anticipation of another view being presented on the video. The observation of 12 percent of responses saying the classes were “Equal” in Part (d) was not seen in the earlier studies. The accompanying reasoning appeared to suggest this response often reflected an inability to decide between the two classes rather than a strong justification of equality. It was more of a qualitative description, for example in predicting what might happen if the two classes had the same number of children in them. Student S8 appeared to reflect both of these views at different times. Some students may also have been influenced by the appropriateness of “Equal” as a response to Part (c). As noted the grade 9 students performed somewhat less well than the grade 9 students in the Watson and Moritz study. Given the likelihood that the grade 9 group in the current study was above average in ability, this could be seen as a disappointing result; it may however reflect a lack of classroom experience on issues involved with comparing data sets in the middle school years. Of particular interest for this study of cognitive conflict was having a distribution of initial responses that would allow for conflict to have an impact, mainly for improved response but also for testing appropriate justifications. For Part (c) the statistically appropriate response of “Equal” – perhaps with qualification – was given by 62 percent of students, whereas for Part (d) only 17 percent chose the “Black” class as better. There were hence fewer opportunities for improvement in Part (c) and many students were not prompted with conflict for alternative responses. 6.2. Research questions 2 and 3 – Conflict for Parts (c) and (d) For Part (c), where the statistically appropriate response of “Equal” was suggested by many students, just over half who had other criteria for choosing the “Yellow” or “Brown” class were convinced by an “Equal” argument. Some had not thought to sum the scores. In retaining arguments favouring “Yellow” for being more consistent (more 5s), or “Brown” for being more evenly spread or for having the highest score, students were using criteria that in some circumstances would be considered viable. It is important, therefore, for teachers to discuss how situations determine

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the criteria used for decision making and if necessary distinguish between what is considered statistically appropriate and what one might do if one wanted the class with the highest score to enter a competition. Students should also be encouraged to consider alternatives such as the response of student S4 in Watson (2001), which canvassed all possibilities noted here. Part (d) was included in the protocol to gauge student reaction to data sets of unequal sizes. The question was set in terms of class size rather than “sample” size to avoid confounding with issues of sample selection or appropriate sample size, as these were dealt with elsewhere in the larger study (Watson and Moritz, 2000). Of interest first was whether students would notice the unequal sizes of the classes and second, if they did, what strategy would be used to handle the situation. No students had difficulty with the inequality of groups, even if it had to be pointed out to them. Of the 83 percent of students who did not choose the “Black” class based on a comparison of means or the visual disproportion in that graph favouring it, only 30 percent were convinced by arguments from the students on video. It may be that lack of opportunity to argue back and forth posed a difficulty but the interviewer often repeated an argument if the interviewee appeared to be confused. The prompts, being from other students, rather than an expert, may not have been optimal (this is almost certainly so) and hence not as convincing as they could have been. There is also evidence associated with the final research question, that the visual “hump” in the graph for the “Pink” class was a strong cue, particularly for younger students. The change to “Black” by about 30 percent of students was slightly higher than the 24 percent change to “Black” observed after interviewer probing on class size by Watson and Moritz (1999) and the 26 percent change found again after probing by Watson (2001). It is interesting, however, to compare the percent of students who changed their minds to “Black” after being presented with cognitive conflict of other students, to the percent who improved from “Pink” to “Black” after three or four years (initial responses before probing). Watson (2001) found that of the 42 students interviewed longitudinally from the original study, 17 percent had several years earlier successfully argued that the “Black” class had done better. All of these students subsequently argued in favour of the “Black” class. Of the remaining 35 students who had initially chosen the “Pink” class, 31 percent chose the “Black” class, with appropriate arguments three or four years later. This is virtually the same outcome as achieved in the current study after three or four minutes of interaction with video of other students. Although one could be much more confident of the stability of the understanding displayed by the students interviewed longitudinally, the outcome from the current study shows promise for en-

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couraging student debate in the classroom. If a digitised video resource such as used in this study were available on computer, it would be possible for a teacher to choose judiciously from arguments, allow for student reaction and expand explanations as required. With reinforcement it might be expected that similar results to the longitudinal study could be confidently retained. In comparison to other protocols used with the students in the current study (Watson and Moritz 2001a, 2001b; Watson, 2002) the improvement rates are similar. The 57 percent improvement for Part (c) compares well with the 60 percent who preferred a more appropriate pictograph representation after prompting by other students. The 30 percent improvement for Part (d) is similar to improvement observed for chance measurement, sampling and prediction from pictographs. These are not independent outcomes as the students were the same for each protocol. There is the suggestion, however, in looking at content, that Part (c) and representing with pictographs were easier tasks than Part (d), chance measurement (which was also based on a proportional argument), sampling and prediction. Certainly Part (c) was more straightforward than Part (d) in the current study. It is possible to expect, then, greater success rates for the use of cognitive conflict, if the tasks are easier and not as much improvement is required to reach an optimal response. More research is needed with independent samples to confirm this hypothesis. 6.3. Research question 4 – Variation before and after conflict Two aspects of students’ appreciation of variation were of interest in the light of previous research. One was associated with a confirmation of previous observations of students’ acknowledgement of variation in the graphs of the data sets. Not only was acknowledging the presence of variation important however but also its relationship to an appropriate decision on which class had done better, particularly in Part (d). The second interest was in whether students would further use or introduce arguments based on variation in supporting or attacking conflicting prompts made by other students on video. Students’ mentioning of variation in the initial parts of their interviews followed the same overall pattern in the current study as in Watson (2001) for both Part (c) and Part (d). For Part (d) in particular it was of interest to observe the association of acknowledgement of variation and success in saying that the “Black” class had done better. Of the 48 students who initially took some notice of variation in the graphs for Part (d), 7, or 15 percent, produced a proportional argument favouring the “Black" class, whereas of the 10 who chose “Black”, these represented 70 percent.

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After being prompted by conflict, the percent continuing to mention variation dropped from 80 percent to 66 percent. This contrasts with the rise from 88 percent to 95 percent among the students interviewed longitudinally in Watson (2001). This is not surprising because some of the students in the current study were likely to have felt it unnecessary to repeat previous arguments made a few moments earlier, whereas after three or four years the longitudinal students were beginning arguments from the start. In the current study after being prompted by conflict, of the 39 who mentioned variation, 12, or 31 percent, correctly chose “Black”. These 12 represented 48 percent of the 25 who argued or continued to argue for “Black” after seeing conflict views. Although taking note of variation is not a good predictor of ultimate success in Part (d), those who are successful are more likely to take note of the shape of the graphs to assist in decision making. Given the emerging interest in variation and distributions, the observations from this research would appear to support encouraging the building of understanding of basic features of graphical representations, such as basic shape, columns, clumps and humps, rather than expecting that means will be understood and employed. As observed elsewhere (e.g., Gal et al., 1990; Mokros and Russell, 1995; Watson and Moritz, 1999) the mean is not naturally used by very many school students in situations where it could be an important representative measure. In this study it was not surprising that no grade 3 students mentioned average but perhaps a little so that grade 9 students referred to it less often than grade 6 students. As might be expected, the use of totals decreased with grade level. The recent work of Konold and Pollatsek (2002), which describes central tendency as the search for signals within noisy processes, offers several useful scenarios for instruction to emphasise the relationship between averages and variation. These include repeated measures, measuring individuals and dichotomous events. The process potentially involves students collecting their own data and experiencing cognitive conflict as concepts are consolidated. 6.4. Methodology Several features of the design of this study warrant attention. The choice of students from a range of grades but of higher ability as judged by their teachers was made to allow for a variety of levels of performance while at the same time not causing stress to students. Ethical requirements meant teachers were asked to select students they felt would enjoy the prospect of engaging in cognitive conflict of the sort employed in the study. All teachers expressed pleasure at the opportunity this offered their higher ability

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students. Although from one point of view it might have appeared better to pick a group of students more likely to provide inadequate responses initially, the above considerations were more important. The methodology employed also provided the opportunity to challenge correct arguments and document their stability. In this study no students were persuaded away from appropriate arguments. A limitation to the study from a design perspective was the fact that all prompts were not used with all students regardless of initial response. Although this process might have provided more information, it may have had drawbacks and was not employed for three reasons. First there was a time constraint with other protocols used during the interview. Second there was the view that using too many prompts would become tedious and students would not continue to give reasoned responses. Third it was often the opinion of the interviewer that an appropriate opinion was so forcefully stated that there was no point in challenging it. This decision was sometimes based not only on the current response but also on responses to previous protocols. These considerations should be taken into account in future research where alternative procedures may be deemed feasible. The prompts from earlier interviews were chosen from those available and reflected actual arguments rather than what might be presented in a text book or by a teacher. Hence as can be seen in Figures 2 and 3, they are not perfect. They do, however, reflect exactly the type of arguments one might expect from students interacting with each other in the classroom. If such comments are considered inadequate by statistics educators then those who see an advantage in placing students with others in their Zone of Proximal Development will need to rethink group work and collaboration. What is missing from this study, however, is the continued dialogue in which students might engage and where students providing lifting prompts might further expand and elucidate their arguments. Although it is possible to video tape such interaction in the classroom (Chick and Watson, 2001) there is no control possible on the kind of cognitive conflict presented. One of the questions for future research, however, is the inclusion of expert adult or teacher prompts, to see if this would have a stronger influence on students accepting alternative views. Also of interest for future research is the issue of transfer after the acceptance of higher level views from other students. 7. C ONCLUSION

This research has continued the study of students’ developing understanding of statistical inference within the context of comparing two data sets

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displayed in graphical form. The variation in the displayed distributions allowed for consideration of the features students felt were important in decision making. The framework for this study however also introduced cognitive conflict in the form of video prompts from other students. These were selected by the interviewer to challenge the views expressed by the interviewee. Depending on the complexity of the comparing task, 30 or 57 percent of students who could improve their responses responded in the appropriate fashion to the prompts. The methodology employed in this study offers promise both for controlled research and for use in the classroom by knowledgeable teachers.

ACKNOWLEDGEMENTS This research was funded by an Australian Research Council grant, No. 79800950. Jonathan Moritz conducted the interviews and set up the spreadsheet environment for analysis, with initial coding of responses. Judith Deans transcribed the interviews. An earlier version of this paper was presented at the second Statistical Reasoning, Thinking and Literacy Forum held in Australia in 2001.

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ematics Education, Research Group of Australia Incorporated, Vol. 2), MERGA, Sydney, pp. 523–530. Watson, J.M. and Moritz, J.B.: 2001b, ‘Development of reasoning associated with pictographs: Representing, interpreting and predicting’, Educational Studies in Mathematics 48, 47–81.

Faculty of Education, University of Tasmania, Private Bag 66, Hobart, Tasmania 7001, Australia, Telephone 61-3-62262570, Fax 61-3-62262569, E-mail: [email protected]

LUIS SALDANHA and PATRICK THOMPSON

CONCEPTIONS OF SAMPLE AND THEIR RELATIONSHIP TO STATISTICAL INFERENCE

ABSTRACT. We distinguish two conceptions of sample and sampling that emerged in the context of a teaching experiment conducted in a high school statistics class. In one conception ‘sample as a quasi-proportional, small-scale version of the population’ is the encompassing image. This conception entails images of repeating the sampling process and an image of variability among its outcomes that supports reasoning about distributions. In contrast, a sample may be viewed simply as ‘a subset of a population’ – an encompassing image devoid of repeated sampling, and of ideas of variability that extend to distribution. We argue that the former conception is a powerful one to target for instruction. KEY WORDS: conceptions, sample, sampling, sampling distributions, statistical inference

1. BACKGROUND

On the basis of empirical evidence Kahneman and Tversky (1972) hypothesized that people often base judgments of the probability that a sample will occur on the degree to which they think the sample “(i) is similar in essential characteristics to its parent population; and (ii) reflects the salient features of the process by which it is generated” (ibid., p. 430). This hypothesis suggests that Kahneman and Tversky’s subjects focused their attention on individual samples. In later research, Kahneman and Tversky (1982) conjectured that people, indeed, tend to take a singular rather than a distributional perspective when making judgments under uncertainty. In the former, one focuses on the causal system that produced the particular outcome and assesses probabilities “by the propensities of the particular case at hand” (ibid., p. 517). In contrast, the distributional perspective relates the case at hand to a sampling schema and views an individual case as “an instance of a class of similar cases, for which relative frequencies of outcomes are known or can be estimated” (ibid., p. 518). Konold (1989) found strong empirical support for Kahneman and Tversky’s (1982) conjecture. He presented compelling evidence that people, when asked questions that are ostensibly about probability, instead think they are being asked to predict with certainty the outcome of an individual trial of an experiment. Konold (ibid.) characterized this orientation, which Educational Studies in Mathematics 51: 257–270, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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he referred to as the outcome approach, as entailing a tendency to base predictions of uncertain outcomes on causal explanations instead of on information obtained from repeating an experiment. Sedlmeier and Gigerenzer (1997) analyzed several decades of research on understanding the effects of sample size in statistical prediction. They argued compellingly that subjects across a diverse spectrum of studies who incorrectly answered tasks involving a distribution of sample statistics may have interpreted task situations and questions as being about individual samples. Recent instructional studies (delMas, 1999; Sedlmeier, 1999) indicated that engagement in carefully designed instructional activities using computer simulations of drawing many samples can help orient students’ attention to collections of sample statistics when making judgments involving samples. However, analyses in these studies did not focus on characterizing students’ evolving conceptions and imagery in relation to their engagement in instruction. Despite the centrality of variability in statistics, students’ understanding of sampling variability and our comprehension of variability’s role as a central organizing idea in statistics instruction has received little research attention (Shaughnessy et al., 1999). Rubin et al. (1991) proposed that a coherent understanding of sampling and inference entails integrating ideas of sample representativeness and sampling variability to reason about distributions. Images of the re-sampling process, however, were not at the foreground of their conceptual analysis. Other conceptual analyses of sampling (Schwartz et al., 1998; Watson and Moritz, 2000) characterized the relationship between population and a randomly selected subset of it in a way that did not entail images of the repeatability of the sampling process nor of the variability that we can expect among sample outcomes. In sum, substantial evidence from research on understanding sampling suggests that students tend to focus on individual samples and statistical summaries of them instead of how collections of sample statistics are distributed. Furthermore, students may tend to predict a sample’s outcome on the basis of causal analyses instead of statistical patterns in a collection of sample outcomes. These orientations are problematic for learning statistical inference because they disable students from considering the relative unusualness of a sampling process’ outcome. Finally, sampling has not been characterized in the literature as a scheme of interrelated ideas entailing repeated random selection, variability, and distribution.

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2. P URPOSE AND METHODS

This study investigated the development of students’ thinking as they participated in instruction designed to support their conceiving sampling as a scheme of interrelated ideas including repeated random selection, variability among sample statistics, and distribution. Twenty-seven 11th - and 12th -grade students, enrolled in a non-AP semester-long statistics course, participated in a 9-session whole-class teaching experiment addressing ideas of sample, sampling distributions, and margins of error. Our aim was to develop epistemological analyses of these ideas (Glasersfeld, 1995; Steffe and Thompson, 2000; Thompson and Saldanha, 2000) – ways of thinking about them that are schematic, imagistic, and dynamic – and hypotheses about their development in relation to students’ engagement in classroom instruction. Three research team members were present in the classroom during all lessons: one author designed and conducted the instruction; the other author observed the instructional sessions and took field notes; a third member operated the video cameras. Students’ understandings were investigated in three ways: by tracing their participation in classroom discussions (all instruction was videotaped), by examining their written work, and by conducting post-experiment individual interviews. Instruction stressed two overarching and related themes: 1) the random selection process can be repeated under similar conditions, and 2) judgments about sampling outcomes can be made on the basis of relative frequency patterns that emerge in collections of outcomes of similar samples.1 These themes were intended to support students’ developing a distributional interpretation of sampling and likelihood. Though an a priori outline of the intended teaching and learning trajectories (Simon, 1995) guided the progress of the teaching experiment, the research team made on-line adjustments to instruction according to what they perceived as important issues that arose for students in each session. The teaching experiment unfolded in three interrelated phases: it began with directed discussions centered on news reports that mentioned data about sampled populations and news reports about populations per se (raising the issue of sampling variability). The experiment then progressed to questions of “what fraction of the time would you expect results like these?” This entailed having students employ, describe the operation of, and explain the results of computer simulations of taking large numbers of samples from various populations with known parameters (see Figure 1). The experiment ended by examining simulation results systematically, with the aim that students see that distributions of sample proportions are

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Figure 1. Part of an instructional activity designed to help students make sense of computer simulations of drawing many random samples from a population. Simulation input (left) and output (right) windows were displayed in the classroom and the instructor posed questions designed to orchestrate reflective discussions about the simulations.

Figure 2. Part of an instructional activity designed to structure students’ investigation of the relationship between sampling distributions and underlying population proportions. Students filled out the table on the left by organizing information (like that shown on the right) generated by computer simulations of drawing many random samples from populations with given proportions.

largely unaffected by underlying population proportions (see Figure 2), but are affected in important ways by sample size.

3. R ESULTS AND DISCUSSION

In this report we move toward elaborating an important distinction between two conceptions of sample and sampling that emerged in the teaching experiment. Our analyses revealed that some students – generally those

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who performed better on the instructional activities and those who were able to hold coherent discourse about the mathematical ideas highlighted in instruction – had developed a multi-tiered scheme of conceptual operations centered around the images of repeatedly sampling from a population, recording a statistic, and tracking the accumulation of statistics as they distribute themselves along a range of possibilities. These images and operations were tightly aligned with those promoted in classroom instructional tasks and discussions. As such, we conjecture that these students’ engagement in the instructional activities played an important role in their developing such a scheme. For instance, we had students practice imagining and describing a coordinated multi-level process that gives rise to sampling distributions (and to the simulations’ results): Level 1: Randomly select items to accumulate a sample of a given size from a population. Record a sample statistic of interest. Level 2: Repeat Level 1 process a large number of times and accumulate a collection of statistics. Level 3: Partition the collection in Level 2 to determine what proportion of statistics lie beyond (below) a given threshold value. In classroom discussions the instructor employed a metaphor designed to help students distinguish and coordinate these different levels. The metaphor entails imagining a collected sample of dichotomous opinions (“yes” or “no”) in Level 1 as a box containing ‘1’s (for “yes”) and ‘0’s (for “no”). It then entails labeling each box with a ‘1’ (or a “0”) if the proportion of its contents is greater (or less) than a given threshold value. In this metaphor, what accumulates in Level 2 is a collection of ‘1’s and ‘0’s, each of which represents a sample whose statistic is greater (less) than the threshold value. At Level 3, the metaphor entails calculating the percent of the Level 2 collection that are ‘1’ or that are ‘0’, depending on the required comparison. The following excerpt illustrates one student’s coherent image of the multi-tiered sampling process, the development of which appeared to have been facilitated by his use of this metaphor. We take this student’s coherent image as an expression of the stable scheme of conceptual operations characterized above. In the excerpt, the student (D) interpreted a sampling simulation’s command and the result of running it as he viewed familiar simulation windows on a computer screen (see Figure 1)2 :

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Ok. It’s asking. . .the question is. . .like “do you like Garth Brooks?”. You’re gonna go out and ask 30 people, it’s gonna ask 30 people 4500 times if they like Garth Brooks. The uh. . .(talks to himself) what’s this? let’s see. . .the actual. . .like the amount of people who actually like Garth Brooks are. . .or 3 out of 10 people actually prefer like Garth Brooks’ music. And uh. . .for the 30. . .when you go out and take one sample of 30 people, the cut off fraction means that if you’re gonna count, you’re gonna count that sample, if like 37% of the 30 people preferred Garth Brooks. And then it’s going to tally up how many of the samples had 37% people that preferred Garth Brooks. So like the answer would be I don’t know, like whatever, 2000 out of 4500 samples had at least 37% of people preferring Garth Brooks. [. . .] How was it that you thought about it that allowed you to keep things straight? [. . .] I just thought of it like . . . I don’t know, I sort of thought of it like how you were saying. Like. . .if the like 1s and the 0s if you ask 30 uh if like 10 of them say they like Garth Brooks – or for every person who likes Garth Brooks you put a 1 down, if they don’t you put a zero. You do that 30 times and you’re gonna get like I don’t know, 15 ones and 15 zeros you add up, you add them up. Then it says the cutoff fraction for each sample is 37% so you have like at least 37% of the. . .like those or. . .30 – if you add it up and divided it by the 30 and it’s at least 37% then you have like another pile of like little papers and you put a one on like the big, the big one for the sample or a zero if it’s less than – if the whole sample is less than 37%. The 1s and 0s I don’t know. . .you said something about like. . .that sort of helped.

A significant feature of student D’s thinking was his ability to clearly distinguish different levels of the resampling processes – never confounding the number of people in a sample with the number of samples taken – while coordinating the various levels into a structured whole. Additionally, and relatedly, student D interpreted the result of the simulation as an amount (percentage) of sample proportions, thus suggesting that he understood that the multi-level process generated a collection of sample proportions.3 Student D’s coherent image contrasts sharply with that of many poorerperforming students who persistently confounded numbers of people in a sample with numbers of samples drawn. The following interview excerpt illustrates one such student’s (M) difficulties in the context of explaining similar computer simulations:

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Segment 1 I: Ok, Suppose that, here’s what I’m gonna do, uhh instead of 4500 samples I’m gonna take uhh, 1000 samples. Everything’s gonna stay the same – sample size is 30, population fraction is 3/10ths, but now were’ just taking 1000 samples. What would you expect the results to be? [. . .] M: Uhh, somewhere around like (short silence), hmm around like 25 to 30% of those 1000 samples. I: Why 25 to 30%? M: Because it’s uhh . . . easier to uhh, I mean I: What are you basing that judgment on? M: Uhh, the actual population percentage, of 30 I: Ok, so you figure it’ll be about 30%, 25 to 30, because the population fraction is 30%? M: Yeah, somewhere close to that. [. . .] Segment 2 I: Alright (runs simulation, result displayed on output screen is “189 of these 1000 repetitions . . .”) M: 2/10ths, 20%. Hmm, it’s still a little less I: So it’s a little less than 20%, right? M: Hmm hmm, huh (seems surprised) [. . .] Segment 3 I: Alright. Suppose that now we, let’s do this, let’s make 2500 samples (changes parameter value in command window). What fraction of those samples, I mean what result would you now expect, for the number of samples that we’re going to get that exceed 37% preferring Garth Brooks? M: About 1/5 of those. [. . .] I: Now, before you would have said “well, 3/10ths of the 2500 samples, the 2500 repetitions” M: Hmm hmm I: Do you still sort of lean that way, that you should get around 3/10ths of the –? M: I think it should, but I don’t understand why it’s not, why it keeps coming out with 1/5th rather than 1/3rd .

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Alright, what is that ‘3/10ths’ 3/10ths of? Uhh, hmm 3/10ths of the entire population Alright, and those are people, right? Hmm hmm (nods) Now, if you took 3/10ths of the 2500 repetitions you’re taking 3/10ths of what? Of the uhh. . .people sampled (chuckles) No, 3/10ths of the samples. Oh. Hmm hmm

Segment 1 of the excerpt suggests that student M expected the simulation to produce an amount (number) of samples and that he expected the percentage of that amount to hover around the sampled population percent (30%). Segment 2 illustrates his surprise at finding the actual percent being 20% of the 1000 samples generated. In segment 3 the student anticipated the same (20%) result for a simulation involving a larger number of samples, but he did not understand why this should be so because his conviction was that the simulation should produce a numerical value close to the sampled population percentage. The remainder of the segment reveals that student M had been interpreting the simulation’s result as a percentage of people sampled rather than as a percentage of samples. During such instructional activities most students experienced great difficulty conceiving the re-sampling process in terms of distinct levels. They would often unwittingly shift from speaking and thinking of a number of people in a sample to a number of samples selected. Their control of the coordination between the various levels of imagery was unstable; from one moment to the next their image of a number of samples (of people) seemed to easily dissolve into an image of a total number of people. These difficulties led many students to misinterpret a simulation’s result as a percentage of people rather than a percentage of sample proportions. This muddling of the different levels of the resampling process, in turn, obstructed their ability to imagine how sample proportions might distribute themselves around the underlying population proportion. A salient consequence of these students’ difficulties in imagining a sampling distribution was their tendency to judge a sample’s representativeness only in relation to the underlying population proportion. Their image of sampling did not entail a sense of variability that extended to ideas of distribution: they understood that a sample statistic’s values vary, but only to the extent that if we were to draw more samples and compute a statistic from them, those values would differ from the ones for the samples already drawn. Thus, judgments about the unusualness of a particular value

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Figure 3. An additive image of sample entails only part-whole relationships. Resemblance between sample and population is not a salient issue. Multiple samples are seen as multiple subsets.

of a statistic were based largely on how they thought the value compared to the underlying population parameter per se, instead of on how it might compare to a clustering of the statistic’s values. On the basis of such characteristics, we conjecture that these students’ encompassing image of sample was additive – that is, in these instructional settings they tended to view a sample simply as a subset of a population and to view multiple samples as multiple subsets. A contrasting image of sample is suggested in the following excerpt of student D explaining the purpose of simulating resampling: D:

I: D:

If like. . .if you represent – if you give it like the split of the population and then you run it through the how – number of samples or whatever it’ll give you the same results as if – because in real life the population like of America actually has a split on whatever, on Pepsi, so it’ll give you the same results as if you actually went out, did a survey with people of that split. Ok, now. What do you mean by “same results”? On any particular survey at all – you’ll get exactly what it –? No, no. Each sample won’t be the same but it’s a. . .it’d be. . .could be close, closer. . .

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What’s the ‘it’ that would be close? If you get. . .if you take a sample. . .then the uh. . .the number of like whatever, the number of ‘yes’s would be close to the actual population split of what it should be. Are you guaranteed that? You’re not guaranteed, but if you do it enough times you can say it’s within like 1 or 2% of error depending upon uh how many times – I think – how many times you did it.

The resemblance between sample and population was clearly foremost in student D’s mind, but his image was of a fuzzy resemblance bound up with ideas of variability and proto-distributional images of a collection of sample proportions. He did not expect a sample to be an exact replica of the sampled population, instead he anticipated that in repeating the sampling process many sample proportions would be ‘more or less’ close to the population proportion. Moreover, student D’s confidence in a sample’s representativeness was based on this anticipated image of how a collection of similar sample proportions might be distributed around the population proportion. We put that student D’s description is consistent with his having conceived a sample as a quasi-proportional mini version of the sampled population, where the ‘quasi-proportionality’ image emerges in anticipating a bounded variety of outcomes, were one to repeat the sampling process. It is often useful to refer to a germinating idea with suggestive terminology; we call this image of sample a multiplicative conception of sample (MCS) because its constitution entails conceptual operations of multiplicative reasoning. An elaboration of multiplicative reasoning (Harel and Confrey, 1994) is beyond the scope of this paper. For the present discussion we draw on Inhelder and Piaget’s (1964) broad characterization of multiplicative reasoning as conceiving an object (quantity) as simultaneously composed of multiple attributes (quantities). For instance, conceiving a proportion involves multiplicative reasoning when it entails comparing two quantities in such a way as to think of the measure of one in terms of the measure of the other (Thompson and Saldanha, in press). An example is when one thinks of percentage as quantifying a part of a whole in terms of the whole. This conception entails keeping both the part and the whole simultaneously in mind and the ability to reciprocally relate and express one in terms of the other. This is different from thinking of measuring a subpart of a whole only in absolute terms. We hypothesize that MCS entails multiplicative operations on several levels: on one level it entails conceiving a relationship of proportional-

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Figure 4. A multiplicative conception of sample entails a quasi-proportionality relationship between sample and population. Multiple samples are seen as multiple, scaled quasi mini-versions of the population.

ity between a sample and a population. On another level, imagining the emergence of a proto-distribution of sample statistics entails structuring statistics as subclusters within the range of an entire collection of statistics. This involves fractional reasoning. Finally, a mature and well articulated image of distribution supports quantifying the expectation of a particular kind of sampling outcome and thus quantifying one’s confidence in a sampling outcome’s representativeness. This entails the operation of juxtaposing the individual sample result against an aggregate of similar sample results to compare the one against the many – an image of simultaneity that is central to multiplicative reasoning.

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4. C ONCLUSION

Though our elaboration of these two images of sample and sampling is empirically grounded, our central aim in presenting it is not to imply that students in our experiment adhered steadfastly to one or the other image. Rather, our aim is to highlight two significantly different conceptions of sample and sampling – perhaps exemplary of extremes in a continuum of students’ conceptions – that provide insight into what may be more or less powerful conceptions to target for instruction. From our perspective, there are two reasons why the distinction between the additive and multiplicative conceptions of sample is significant. First, in contrast to the additive image, MCS entails a rich network of interrelated images that supports building a deep understanding of statistical inference. In practice, statistical inferences about a population are typically made on the basis of information obtained from a single sample randomly drawn from the population. This practice is common among statisticians despite expectations of variability among sampling outcomes. In statistics instruction, however, it is uncommon to help students conceive of samples and sampling in ways that support their developing coherent understandings of why statisticians have confidence in this practice. We claim that MCS empowers students to understand the why by orienting them to relate individual sample outcomes to distributions of a class of similar outcomes. In the same way, MCS enables students to consider a sampling outcome’s relative unusualness. As such, we propose that MCS characterizes a powerful and enabling conception to target for instruction; it can guide efforts to design instructional activities and student engagements intended to support their developing a deep understanding of sampling and inference. The second reason why we consider the distinction between these two conceptions of sample to be significant is that few of our students developed MCS. Instead, most students seemed oriented toward an additive image of sample. To us, this state of affairs suggests that developing MCS is non-trivial. The reasons for students’ difficulties in this regard are currently unclear to us. However, one plausible hypothesis grounded in our data is that for many students the simulation and sampling distribution activities were of such a complexity so as to essentially overshadow ideas of sampling variability highlighted in the first phase of the teaching experiment. In a subsequent teaching experiment (Saldanha and Thompson, 2001) we followed this hypothesis and engaged students in instructional activities designed to foreground ideas of sampling variability and support their developing a MCS.

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ACKNOWLEDGEMENT Research reported in this paper was supported by National Science Foundation Grant No. REC-9811879. Any conclusions or recommendations stated here are those of the authors and do not necessarily reflect official positions of NSF.

N OTES 1. Similar samples share a common size, selection method, and parent population. Furthermore, they are selected to obtain information about a common population characteristic. 2. The simulation was of sampling people’s preference for a particular musician from a hypothetical population having a known proportion of it preferring the musician. 3. We note that student D’s prediction of the simulation result was highly inaccurate in this excerpt. Shortly thereafter, however, he quickly revised his prediction with a highly accurate one and continued to make such accurate predictions throughout the rest of the interview. We thus believe that his initial prediction was not an indication of a poor sense of how the sample proportions were distributed, rather it was merely the result of his focus, in the moment, on explaining how the simulation worked and what it generated.

R EFERENCES delMas, R.C., Garfield, J. and Chance, B.L.: 1999, Exploring the Role of Computer Simulations in Developing Understanding of Sampling Distributions, Paper presented at the American Educational Research Association, Montreal. Glasersfeld, E. v.: 1995, Radical Constructivism: A Way of Knowing and Learning, Falmer Press, London. Harel, G. and Confrey, J. (eds.): 1994, The Development of Multiplicative Reasoning in the Learning of Mathematics, SUNY Press, Albany, NY. Inhelder, B. and Piaget, J.: 1964, The Early Growth of Logic in the Child: Classification and Seriation, W.W. Norton and Company Inc., New York. Kahneman, D. and Tversky, A.: 1982, ‘Variants of uncertainty,’ in D. Kahneman, P. Slovic and A. Tversky (eds.), Judgment under Uncertainty: Heuristics and Biases, Cambridge University press, New York, pp. 509–521. Kahneman, D. and Tversky, A.: 1972, ‘Subjective probability: A judgement of representativeness,’ Cognitive Psychology 3, 430–454. Konold, C.: 1989, ‘Informal conceptions of probability,’ Cognition and Instruction 6(1), 59–98. Rubin, A., Bruce, B. and Tenney, Y.: 1991, Learning about Sampling: Trouble at the Core of Statistics, In D. Vere-Jones (ed.), Proceedings of the Third International Conference on Teaching Statistics, ISI Publications in Statistical Education, Dunedin, New Zealand, Vol. 1, pp. 314–319.

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Saldanha, L.A. and Thompson, P.W.: 2001, ‘Students’ reasoning about sampling distributions and statistical inference,’ in R. Speiser and C. Maher (eds.), Proceedings of the Twenty Third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, ERIC Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, OH, Snowbird, Utah, Vol. 1, pp. 449–454. Schwartz, D.L., Goldman, S.R., Vye, N.J. and Barron, B.J.: 1998, ‘Aligning everyday and mathematical reasoning: The case of sampling assumptions,’ in S.P. Lajoie (ed.), Reflections on Statistics: Learning, Teaching, and Assessment in Grades K-12, Lawrence Erlbaum Associates, Mahwah, NJ, pp. 233–273. Sedlmeier, P.: 1999, Improving Statistical Reasoning: Theoretical Models and Practical Implications, Lawrence Erlbaum, Mahwah, NJ. Sedlmeier, P. and Gigerenzer, G.: 1997, ‘Intuitions about sample size: The empirical law of large numbers,’ Journal of Behavioral Decision Making 10, 33–51. Shaugnessy, J.M., Watson, J., Moritz, J. and Reading, C.: 1999, School Students’ Acknowledgment of Statistical Variation, Paper presented at the Research Presession Symposium of the 77th Annual NCTM Conference, San Francisco, CA. Simon, M.A.: 1995, ‘Reconstructing mathematics pedagogy from a constructivist perspective,’ Journal for Research in Mathematics Education 26(2), 114–145. Steffe, L.P. and Thompson, P.W.: 2000, ‘Teaching experiment methodology: Underlying principles and essential elements,’ in A.E. Kelly and R.A. Lesh (eds.), Handbook of Research Design in Mathematics and Science Education, Lawrence Erlbaum Associates, inc., Mahwah, NJ, pp. 267–306. Thompson, P.W. and Saldanha, L.A.: in press, ‘Fractions and multiplicative reasoning,’ in J. Kilpatrick and G. Martin (eds.), Research Companion to the Principles and Standards for School Mathematics, NCTM, Reston, VA, pp. 95–114. Thompson, P.W. and Saldanha, L.A.: 2000, ‘Epistemological analyses of mathematical ideas: A research methodology,’ in M.L. Fernandez (ed.), Proceedings of the Twenty Second Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, ERIC Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, OH, Tucson, AZ, Vol. 2, pp. 403–408. Watson, J.M. and Moritz, J.B.: 2000, ‘Developing concepts of sampling,’ Journal for Research in Mathematics Education 31(1), 44–70.

Vanderbilt University, Dept. of Teaching and Learning, GPC Box 330, 240 Wyatt Center, Nashville, TN 37203, Telephone (615) 322-8100, Fax (615) 322-8999, E-mail: [email protected]

BOOK REVIEW

Pessia Tsamir

Learning and Teaching Number Theory: Research in Cognition and Instruction by S.R. Campbell and R. Zazkis (eds.) Ablex Publishing, ISBN 156750 6534, November 2001, 245 pp When starting to write this book review, I asked myself – what would a potential reader of this review, a mathematics educator involved in teacher training or in teaching postgraduates, or a mathematics education researcher, expect to find in my review? When reading such reviews, what am I, as a teacher educator and a researcher, interested in? I believe that one answer is – I would like to get enough information to know whether or not to read the book under review. I found the task of conveying this message quite a responsibility. On the one hand, this review should endow the reader with a good sense of what this monograph entails, and to enable him/her make their own independent decisions about it. On the other hand, the review should offer a warranted critique of the strengths and weaknesses of the monograph and possibly a concluding opinion. To answer this twofold demand, my review includes descriptive and critical summaries of both the general picture created by the monograph as a whole, and of each chapter as an independent unit. Before going into details, I would like to state that my bottom line is that this monograph would be of great interest to both the teacher educator and to the mathematics education researcher as representatives of the mathematics education community, and to leading mathematics teachers in elementary, as well as secondary schools. This monograph offers a unique opportunity to address a rich variety of aspects dealing with learning and teaching of number theory. It is a well chosen and carefully designed collection of articles, revolving around the investigation of ways to promote prospective teachers’ and college students’ understanding of number theory, and around the examination of their related performance during and after participation in relevant courses. Number theory is presented both as an important mathematical subject area and a valuable, ‘friendly’ field for examining and promoting general mathematical skills like conjecturing, generalizing, proving and refuting mathematical statements. The editors Educational Studies in Mathematics 51: 271–286, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Campbell and Zazkis, while highlighting the values of number theory, call attention to and protest its being a rather neglected area in mathematics education research. They thus present the collection of articles in the monograph as an initial attempt to fill the void, and it is this original, novel nature of their contribution, that grants this collection extra strength. Clearly, the editors achieved their explicitly stated goal “to identify and demonstrate some of the different kinds of problems and ways of thinking that can be investigated in a program of research into learning and teaching number theory and its implication for cognition and instruction” (p. 2). And even beyond that, as Selden and Selden explain, they have focused “on questions and new directions for investigation, some ranging well beyond number theory itself” (p. 214). This review first relates to the outline of the monograph by means of an overview of the different types of the chapters included, and a brief description of the special contribution of each chapter to the global picture of learning and teaching of number theory. In conclusion, I call attention to several motives related to wide-ranging issues regarding cognition and instruction, which recur in the monograph. An overview The monograph consists of eleven chapters that can be classified in various ways, some of which are suggested by the editors in chapter 1 (p. 9). My categorization of the articles is done with reference to the monograph’s title – that is, by focusing on aspects of cognition and instruction. While to some extent all articles in the monograph are concerned with cognition and / or instruction related to number theory, the varying weight given to these two aspects in the different articles sets up the spectrum of this monograph. The first and the last chapter frame the monograph by making explicit its aims and discussing the role that number theory can and should play in Kpost secondary mathematics curricula. They also provide a general survey of the scope of research presented in the monograph, some theoretical underpinnings, as well as additional research questions and suggestions for possible educational implications. The nine remaining chapters discuss issues related to the learning and teaching of number theory, and they can be grouped in the following manner: Four chapters are devoted mainly to students’ conceptions, images, difficulties, and linguistic imprecision when dealing with elementary number theory issues (addressing prospective teachers in chapter 2, 3, 4, and undergraduate computer science students in chapter 5). Three chapters focus on didactical considerations regarding the design of instruction and teaching elementary and advanced number theory problems to prospective

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teachers (discussing general criteria for the choice of tasks to be presented in class in chapter 6, didactical advantages of using generic proofs in chapter 9, and analyzing a specific, recommended task in chapter 7). The two remaining chapters are mainly concerned with connections between the teaching and the learning of number theory, with reference to the role of conjectures and induction proofs in class (conjectures in chapter 8, and induction in chapter 10). It should be noted that by covering various dimensions of learning and teaching number theory, the collection of articles in this monograph provides a significant amount of information regarding cognition and instruction of undergraduates and prospective teachers. Still, each of the nine chapters presents a specific, independent case that is interesting in itself. The following section will allow a glance at their contribution.

A brief summary of the distinct contribution of each chapter The brief summaries of the different chapters will be sequenced according to their presentation in the monograph. As mentioned before, in Chapter 1, “Toward Number Theory as a Conceptual Field”, the editors of the monograph, Stephen Campbell and Rina Zazkis present their aims in editing this book. They, then, concisely discuss the main concerns of number theory as a mathematical subject area, characterize elementary number theory as distinguished from advanced number theory, and briefly review the role of elementary number theory in the K-12 and post-secondary curricula with reference to the explicit and implicit ways in which teaching and learning of number theory are addressed in the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) and in the new Principles and Standards (NCTM, 2000). At this point, Campbell and Zazkis move on to shed some light on the articles included in the monograph. They start by discussing the studies’ constructivist orientation and the theoretical frameworks (e.g., Vergnaud’s theory of conceptual fields, and Dubinsky’s Action-Process-Object-Scheme (APOS) theory) that are used to analyze the data. Then they define number theory as a conceptual field (NTCF) and suggest how the articles in the monograph contribute to this emerging area of research in mathematics education. This opening chapter is not only a good introduction to the monograph; it is also valuable for the description of the possible contribution of number theory issues to learners’ mathematical knowledge of whole numbers and operations, and their suggestions for further examination of this area.

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In Chapter 2, “Coming to Terms with Division: Preservice Teachers’ Understanding”, Campbell discusses preservice teachers’ understanding of the differences between whole numbers division and rational number division, and the difficulties they encountered in coming to terms with these two mathematical situations. Campbell interviewed twenty-one volunteer preservice elementary teachers enrolled in a course that covered basic topics from elementary number theory, and in this chapter he analyzes their correct, as well as their incorrect, ideas when solving the different tasks. The findings indicate that, while the participants’ correct responses are commonly based on calculations, and rarely on divisibility criteria, the basis for their erroneous solutions vary. For example, prospective teachers encountered difficulties with the demand that in whole number division the remainder and quotient are whole numbers (arriving at the solution 0.5 when asked about the remainder of 21 divided by 2). They also encountered difficulties with the demand that the remainder be smaller than the divisor (arriving at the solution the quotient is 9 and the remainder 3 for 21 divided by 2, because 2×9 = 18 and 18+3 = 21) (p. 20–21). The chapter suggests some reasons for the apparent mistakes, relating incorrect responses mainly to participants’ inability to distinguish between rational number division and whole number division with a remainder, to their tendency to inappropriately extend the applicability of familiar processes that are correctly used in certain activities to novel situations, and on their overdependence on interpreting formal referents of arithmetic division using informal language (p. 36). It concludes by raising some related questions for further research. In Chapter 3, “Conceptions of Divisibility: Success and Understanding”, Anne Brown, Karen Thomas and Georgia Tolias report on a study that examined the understanding of ten prospective elementary teachers, enrolled in a course that addressed issues drawn from the number theoretic mathematical contexts regarding basic concepts of introductory topics in number-theory. The authors are interested in the individual’s ability to progress from action-oriented responses to explicit inferential reasoning that may reflect an understanding of mathematical operations and properties concerning the multiplicative structure of the set of natural numbers (a la Freudenthal). The shifts in the participants’ reasoning are illustrated and described as going from stage 1, i.e., performing actions successfully with little awareness of general mechanisms (knowing by doing); to stage 2, where actions and conceptualization influence each other but the individual still cannot make inferences about the success or failure of actions without actually carrying them out; and finally, to stage 3, when actions are consciously

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guided by and reasoned about through applying one’s conceptualization of the task – work in this stage includes the ability to make predictions of the success of future actions without direct experimentation. The authors’ analysis of the data is done by means of a theoretical framework that combines the APOS theory with a stage model adapted from Piaget’s work in Success and Understanding, and the discussion includes a description of previous, related publications by Zazkis and Campbell (1996a, 1996b), where the development of the divisibility concept is analyzed. The authors mention some advantages of the stage model for analyzing an individual’s grasp of divisibility, and suggest that an awareness of the roles played by the various aspects of multiplicative structure is an essential first step in developing the coherence necessary to have a schema for divisibility (p. 46–48). The chapter concludes with a comprehensive discussion of the findings in light of the theoretical framework and of previous publications. Special emphasis is put on detailed pedagogical suggestions referring to a number of number-theory oriented didactical conclusions (e.g., to emphasize the central role of multiplication, or to make links across LCMs different representations), and some general conclusions regarding students’ mathematical behavior. The authors’ last statement raises the pressing need of more research to extend the related, existing body of knowledge. In Chapter 4, “Language of Number Theory: Metaphor and Rigor” Zazkis addresses the increasing awareness of the mathematics education community to the importance of communication and discussion in mathematics classes, and calls attention to the crucial role that precision should play in students’ language when expressing mathematical ideas. Zazkis examines the terminology used by prospective elementary teachers, enrolled in a course called “Principles of Mathematics for Teachers”, when referring to the notion of divisibility during individual interviews. Her findings indicate that most participants use a mixture of informal and formal mathematical terminology. The language used by the interviewers during their interviews was considerably different from the language used by the interviewees, even though the five mathematically equivalent statements that express the notion of divisibility were familiar to the participants from their textbook and from class discussions. Prospective teachers claimed, for instance, that a number ‘cannot be divided’ or ‘cannot be divided evenly’ by 2, meaning that it was not divisible by 2, and occasionally went even as far as to invent words like timesing saying, “You’re timesing it by 6, it’s a multiple of 6”. Zazkis also shows that the terminology used by the participants may reflect their grasp of division. That is, prospective teachers’ saying that

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a number ‘can be divided evenly’ reflects images and processes consistent with a partitive view of division, and their saying that a number ‘goes into’, ‘fits into’, or ‘can be put into’ another number points to their quotitive or measurement view on division. Moreover, several prospective teachers mention that the result is ‘without remainder’, pointing to thoughts about whole number division, whereas other prospective teachers mention division ‘with no decimals’ indicating reference to rational number division. Zazkis identifies possible reasons for the participants’ application of informal terminology. Some suggested reasons refer to the learner e.g., the prospective teachers’ need to confirm for themselves the meaning of the word intended by the interviewer. Other reasons refer to the structure of the word ‘divisible’ – suggesting that like other verbs using the suffix ‘-able’ or ‘-ible’, e.g., ‘edible’ means ‘can be eaten’, it is reasonable to interpret ‘divisible’ as ‘can be divided’. All in all, the analysis of the participants’ responses and their descriptions of divisibility led to the identification of four different, though not necessarily disjoint, themes in application of informal vocabulary: (a) attempting to interpret the word divisible, (b) invoking images and processes, (c) seeking confirmation of meaning, and (d) overemphasizing. In Chapter 5 “Understanding Elementary Number Theory at the Undergraduate Level: A Semiotic Approach”, Pier Ferrari focuses difficulties experienced by Italian first-year university computer science students enrolled in an introductory modern algebra course when dealing with elementary number theory. His study was motivated by four main goals: (a) to discuss undergraduates’ performances when dealing with elementary number theory problems, particularly emphasizing impredicative problems; (b) to test the notion of semiotic control; (c) to test the APOS framework with a different population and to relate it to semiotic control; (d) to begin an analysis of the influence of language and format in the statement of problems on students’ performances. The findings indicate students’ tendency to be “procedural rather than conceptual”, for instance, when linking between division and prime decomposition (p. 110), and the extreme difficulties they encounter in solving rather simple tasks that lack a well-known solving algorithm. The notion of ‘semiotic control’ and the author’s suggested criteria for evaluating students’ behaviors proved useful when interpreting the prospective teachers’ behaviors. Among the criteria mentioned are students’ ability to judge and consider the applicability of different strategies of solutions, and their ability to work with different representations of the same concept. It was

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found, for instance, that participants faced major difficulties in linking divisibility to factorization. These data suggest that Dubinsky’s APOS scheme and Zazkis and Campbell’s analysis of the development of divisibility concepts also apply to undergraduate computer science students. In a sense, ‘semiotic control’ may enrich the APOS framework, as it allows one to analyze behaviors with respect to the interpretation of statements, providing tools to detect and analyze difficulties caused by poor mastery of language. The author concludes by stating that elementary number theory has proved a subject suitable for analyzing undergraduates’ semiotic control of their behavior, and then describing some pedagogical opportunities elementary number theory affords at the undergraduate level for the development of advanced mathematical thinking. In Chapter 6, “Integrating Content and Process in Classroom Mathematics”, Ann Teppo brings a fresh didactical breeze to the monograph that up until this point has focused on clinical studies, mainly examining participants’ cognitive understanding of subject matter issues. This chapter provides a detailed description of an interesting example for how teaching number theory can be facilitated by a didactical approach of ‘reflective discourse’. It describes “a classroom activity based on ideas of number theory that successfully integrates content and processes in the spirit of the new Principles and Standards” (p. 118). By focusing on one 50 minutes lesson, analyzing the mathematical tasks presented, some teaching approaches (e.g., using the ‘empty chart’) and the classroom vignette, Teppo invites us to join her in leading her students on a journey towards the formulation of new sociomathematical norms (a la Yackel and Cobb, 1996) causing them to take part and “become involved in a new type of classroom mathematics” (p. 118). Teppo shows how the prospective teachers who entered her course believing that mathematics is basically a procedurally oriented subject and that studying mathematics is about memorizing formulas and rules gradually became more and more engaged in a wide range of mathematics processes, including organizing information, looking for numerical patterns in order to make generalizations, raising and testing conjectures about these generalizations, and forming abstractions. These processes served for discussing and formulating number theory notions such as factorization, divisibility, and prime and composite numbers. In designing and analyzing her didactical plans, Teppo pays much attention to the mathematical content as well as various aspects of communication, i.e., the importance of students’ expressing ideas and critically listening to their peers, using precise terminology, making conjectures, de-

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fending ideas and justifying their choices. In conclusion, Teppo illustrates didactic tools and approaches which teachers, who aim to motivate and create an atmosphere of inquiry in their classes, can find useful. In Chapter 7, “Patterns of Thoughts and Prime Factorization”, Anne Brown presents a “thought-provoking problem” that, at least at first glance, seems quite difficult and very perplexing. In her phrasing: “though the problem is elementary, no one found it completely trivial or obvious, and its solution elicited a variety of strategies. The strategies as well as the stumbling blocks provide a few insights into the subtleties of the use of prime factorization as tools for reasoning about multiplicative reasoning” (p. 133). The chapter illustrates a way in which a specific representation of a problem, i.e., the prime decomposition of the elements of a given sequence, triggers a wide range of mathematical issues, ideas and difficulties to be clarified and discussed in class. Brown opens the chapter by giving six entries in a sequence, all in prime-factored form, and then challenging the reader to pause, before going on with his / her reading, and solve the following task: (a) Write the next six entries in the sequence, all in prime-factored form, (b) Write the 200th term in prime-factored form, and (c) Describe a method that will provide the prime factorization of the nth term of the sequence. Then, she describes the origins of the problem and explains that by means of this (and similar) problems it is possible to trigger the examination of prime factorization to identify, compare, and contrast the multiplicative properties of natural numbers. She continues by presenting a number of strategies for related solutions, accompanied by enlightening explanations and comments regarding students’ preferences and difficulties. She concludes by raising and discussing some related pedagogical issues, and ‘the interested reader’ is then offered some additional, similar problems for further thought. In Chapter 8, “What Do Students Do with Conjectures? Preservice Teachers’ Generalizations on a Number Theory Task”, Edwards and Zazkis discuss some didactical considerations that underlie the design and the presentation of the Diagonals in a Rectangle, a generalization task. The task is special in the sense that it is embedded in a geometrical setting, but its solution requires some number theory considerations such as the idea of GCD. The authors describe their (as teachers) follow up process on their students’ performance on the task by means of the ‘problem solving journal’ in which the prospective teachers (as students) were asked to record all of their attempts to solve the problem. In addition to the wide instructional background, attention is paid in this chapter to the nature of the prospect-

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ive teachers’ solutions, their problem-solving strategies, and their specific conjectures about possible rules or formulas. The latter, the conjecturingprocess, is described by the authors as follows: “it is this part of ‘preproof’ process that we were most interested in exploring in the research” (p. 141). Indeed, the findings of this study contribute significantly to reported data on the range of prospective teachers’ responses to disconfirming evidence when exploring conjectures. Here, information is gathered when relating to a non conventional number theory problem. The findings indicate, for instance, that most of the participants react appropriately the first time they encounter evidence that does not confirm their conjectures. Still, a few participants do not give up their conjectures due to the negating evidence; they either choose to ignore the evidence or focus on the part of their solution that is consistent with their initial claims. In Chapter 9, “Generic Proof in Number Theory”, Rowland opens by stating his view that “the potential of the generic example as a didactic tool is virtually unrecognized and unexploited in the teaching of number theory”. He is, thus, “urging a change in this state of affairs” (p. 157). Accordingly, this chapter is dedicated to highlighting and discussing the pedagogical advantages of ‘generic proofs’ and to the place the author believes they should have in number theory courses. Rowland first discusses the ‘conviction, explanation and illumination’ purposes of proofs in mathematics classes, and, while referring to published views (e.g., Reuben Hersh and Gila Hanna), states that “In the teaching context, the primary purpose of proof is to explain, to illuminate why something is the case rather than to be assured that it is the case” (p. 159). The author, then, illustrates and discusses the teacher’s role in promoting students’ inductive inference as well as their deductive reasoning, explaining that, naturally, the teacher’s decisions regarding what proofs might be acceptable in a given context, is to a great extent dependent on his / her purposes in ‘proving’. Then, the author presents and analyzes a very rich and stimulating collection of number theoretic examples, some from the literature and others from his own experience as a teacher, thereby illustrating how generic examples might point to general arguments. This collection includes reference to ‘the summation of consecutive odd numbers’, ‘Gauss and the sum 1+2+3+. . .+100’, ‘Euler’s -function’, and to ‘Wilson’s theorem’. Rowland, then, relates to the need for a list of principles underpinning the construction and presentation of generic proofs in number theory, and while he understands that it is premature to offer a definitive list, he makes a first step in this direction by suggesting five guiding principles. The author,

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then, discusses his related work with undergraduate students, describing their obstacles when addressing number theory statements. In the last section of this article ‘pedagogical suggestions and proposals for further research’, Rowland challenges his initial, extreme position of questioning the necessity of formal proofs in general symbolic notation. He takes a ‘more moderate stance’, making ‘three modest and conservative suggestions’, based on his five principles, providing support for students bridging the gap between generic understanding and general exposition (writing ‘proper’ proofs) (p. 180). In Chapter 10, “The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction”, Harel addresses relationships between the learning and teaching of mathematical induction – a significant proof technique in discrete mathematics and in number theory that can provide a context to enhance students’ conceptions of proof. The author explores students’ conceptions and difficulties with mathematical induction in a standard instructional treatment, and in an alternative instructional system. He points to major deficiencies of the standard treatment of mathematics induction: It is handed to students as a prescription to follow; problems can be solved by means of mathematical induction with little understanding of it; and its presentation consists of sequenced problems, from easy to difficult in the view of the author / teacher, rather than in accordance to students’ conceptual development. The alternative treatment of mathematics induction took into account these deficiencies. That is to say, the novel treatment considered the possible causes for failure of the standard approach by including phases corresponding to the levels of conceptual development. It was implemented in a teaching elementary number theory experiment that was carried out with twenty-five junior prospective secondary teachers. The DNR system of pedagogical principles – the duality principle, necessity principle, and repeated reasoning principle for designing, developing and implementing mathematics curricula – is the conceptual basis for this instructional treatment. Harel specifies that the most significant result in this study is “that in this alternative treatment students changed their current ways of thinking, primarily from mere empirical reasoning. . . into transformational reasoning” (p. 206). That is to say, the new pedagogical approach helped students in developing their proof schemes. This chapter describes a most structured study about the pedagogical aspects of designing and carrying out the teaching of number theoretic topics, while emphasizing issues of mathematical induction.

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Finally, in Chapter 11, “Reflection on Mathematics Education Research Questions in Elementary Number Theory”, Annie Selden and John Selden provide an inclusive overview of various general issues that are highlighted in the monograph. They identify the following common threads running through the various articles: “the potential of number theory for teaching and learning of problem solving, reasoning and proof; questions regarding the language and images of divisibility; philosophical stances taken; theoretical frameworks used; and implications for teaching” (p. 214). The authors interweave their discussions with some enlightening comments to which they add illustrations from their own classes, research and readings, and refer to points that need further clarification by means of additional research. In their own words, “in the spirit of the monograph and drawing on it, we focus primarily on questions and new directions for investigation, some ranging well beyond number theory itself” (p. 214). To illustrate, let us look at their discussion of the first topic, i.e., the role elementary number theory could play in promoting students’ mathematical reasoning, generalization, abstraction and proof. The authors first refer to the related statements made in the NCTM Principles and Standards (2000) and to related chapters in this monograph (chapter 6, 9 and 10), describing the chapters’ contribution of data related to the issues under consideration and posing a number of questions for further research (e.g., regarding the effectiveness of generic proof for teaching). The authors go on by describing their own teaching experience of abstract algebra to prospective secondary teachers. They indicate that while proofs are an integral part of abstract algebra courses, in these courses students often find themselves struggling with both, issues related to abstraction and with the constructions of proof. “In contrast, in elementary number theory, students deal with objects (integers) and operations (ordinary multiplication and addition) that are familiar to them. Hence, they can concentrate on discovering and constructing proofs without being distracted by simultaneously having to extend their conceptions of the operations and objects they are studying” (p. 215). In this spirit, the authors suggest that, in order to promote students’ performance with mathematical proofs, elementary number theory statements not involving excessive abstraction be presented. The authors illustrate such a statement, and describe how they have successfully tried it with their university students in a ‘bridge’ course. They conclude by suggesting three major reasons why number theory is ideal for introducing students to reasoning and proof. Similar comprehensive discussions are presented with reference to each of the listed topics.

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This chapter is a special contribution to the monograph, demonstrating via an examination of the other papers and an extended discussion, how number theory lends itself to promoting teaching and learning of general mathematical issues.

Closing Circles and Opening New Horizons I have two reasons for labeling this section: Closing Circles and Opening New Horizons. First, in this section I intend to close circles I have opened in the introduction to this review by offering teacher educators and mathematics education researchers some warranted recommendations regarding this monograph, and by suggesting new horizons of work (teaching and researching) that may evolve from their reading. The other reason for this title is the nature of closing circles and opening horizons I have found in the monograph itself. While addressing various issues related to cognition and instruction of number theory, topics that are the focus of one chapter, time and again, are backed up by the findings reported in other chapters. Moreover, the circles of cognition and instruction are firmly linked and the role each of these plays in the different chapters creates a puzzle worth solving. On the other hand, in the words of Selden and Selden, this monograph “intentionally raises more questions than it answers”, aiming to convince the reader that “many interesting questions in the teaching and learning of number theory await their attention” (p. 213) and thus, the reading of this monograph, may open new horizons of interests and deeds, in teaching and investigating students’ performance with number theory using various types of instruction. I opened this review by asking – what message does this review carry to the teacher educator, and to the mathematics education researcher? Do I believe that they will find any interest in reading this monograph? If the answer is, yes, in what respect? and Why? Before going into details, the answer to both is: YES. I do see many varied benefits for each of them in reading this monograph, and I have also provided them with the flavor of each chapter via a concise summary, so that they will be able to get their own ideas regarding the different issues dealt with here. Now, I would like to close the circle by separately addressing teacher educators, and mathematics education researchers, who are assumed to have read this review up to this point, and to have acquired a sense of moderate familiarity with the various chapters. I also suppose that both are convinced by now that “number theory offers many rich opportunities for explorations that are interesting, enjoyable and useful. These explorations have payoffs in problem solving, in understanding. . . other mathematical

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concepts, [and] in illustrating the beauty of mathematics” (NCTM, 1989, p. 91). First, the teacher educator for elementary (and perhaps even secondary) mathematics teachers, is going to find many ideas on how to teach number theory as well as how to use elementary number theoretic tasks as a rich setting for promoting his students’ mathematical reasoning, problem solving and mathematical communication. As specified in the Overview section, the articles in the monograph are concerned with either cognition or instruction both with regard to number theory. If one is eager to start by getting ideas regarding ways to present number theory to students, (s)he would start by thoroughly reading chapters 6, 7, and 9, which are instruction oriented. One may then continue by reading chapters 8, and 10 that give balanced attention to cognition and instruction, in order to become familiar with both ideas for teaching and a notion of how students may react to such instructional steps. Still, since teaching should consider what students find easy, what they find difficult and why, a teacher educator is likely to find interest in chapters 2, 3, 4, and 5 that focus on students’ conceptions, reasoning, and difficulties. On the other hand, perhaps teacher educator should begin this adventure by reading the editors’ introductory chapter (1), and to conclude by getting some more ideas, and general perspectives, by reading Selden and Selden’s chapter (11). All in all, this monograph explicitly (as the main focus of certain chapters), and implicitly (in the background of some chapters), offers a spectrum of instructional ideas, approaches and tasks. The tasks are presented either as teaching or as research tools – appearing in the introduction to the chapters, or in their description of the study, or as suggestions for additional related problems. They address various mathematical issues, including: (a) elementary number theoretic notions such as, divisibility (e.g., in chapter 2, 3, 4, 5, 7), prime decomposition (e.g., in chapter 2, 5, 6, 11), LCM – least common multiple (e.g., in chapter 3, 7), GCD – greatest common divisor (e.g., chapter 5, 8), and connections between equivalent expressions for the same notion as well as distinction between non-equivalent notions (e.g., in chapter 3, 4). (b) Advanced number theoretic issues, such as the division algorithm/ theorem (e.g., in chapter 2), and Wilson’s theorem (in chapter 9). (c) Additional mathematical settings, like sequences (e.g., in chapter, 7, 10) and rectangular figures (in chapter 8). The reader is usually provided with satisfactory mathematical highlights and didactical comments that point out the special offering of the different tasks, their solutions, and

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occasionally, students’ prevailing ideas and difficulties when solving the tasks. The problems are presented in various ways, including numeric, parametric, (e.g., in chapter 5) and verbal (chapter 2, 3) representations of ‘solve’ tasks (with or without calculators); ‘reversed’ tasks – providing the ‘solution’ while asking about the given (e.g., in chapter 3, 5); ‘generalize’ tasks (e.g., in chapter 6) and ‘prove’ tasks (e.g., in chapter 9, 10, 11). The numeric representations included tasks with whole numbers (e.g., chapters 2, 3), prime factorizations of whole numbers (e.g., 33·52·7 in chapter 2, 3), expressions that include multiplication and addition (e.g., 6·147+2 in chapter 2), sequences and others. The sequencing of these tasks during interviews and in teaching sessions is occasionally discussed and elucidated by the authors. As mentioned in the previous section, additional didactical issues are presented, for instance, in the following chapters: Rowland (chapter 9), convincingly argues for the use of generic proofs for advanced number theory theorems, and suggests five principles for selecting particular cases. Selden and Selden (chapter 11) further explain why and how elementary number theory should serve to promote students’ ability to cope with formal proofs. Harel (chapter 10) shows that the sequencing of tasks should be consistent with students’ conceptual development, and that in the case of mathematical induction, starting with challenging tasks is more beneficial than going ‘from easy to difficult’. Teppo (chapter 6) describes in detail her instructional steps, addressing individuals, small groups, and then, the whole class, using various approaches and tools (e.g., the divisor table), to trigger inquiry, conjectures, justifications, and genuine communication. It is noticeable that I decided not to bring examples of the tasks, and I should emphasize that in no way do I claim to have exhausted the richness of tasks and didactical offerings of this monograph. I left for the reader much to reveal by himself / herself when reading the monograph. Clearly, I believe that the mathematics teacher educators may enjoy and can benefit from reading the monograph. Now, what about the mathematics education researchers? What interest will they find in this monograph? A mathematics education researcher is going to learn a lot about the initial research steps made in “working towards a systematic definition of number theory as a conceptual field” (p. 8). For example, the data reported here, regarding university students’ difficulties when solving elementary number theory tasks, and their use of vague terminology (e.g., in chapter 5, 6), their confusing whole number with rational number division (e.g., in chapter 2), their problems in raising conjectures validating statements and proving (e.g., in chapter 8, 9, 10), and possible reasons for

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these difficulties, should be studied carefully when designing continued studies. Also, the didactical approaches reported, occasionally with no accompanying research (e.g., in chapter 7), and the additional, pedagogical suggestions made (e.g., in chapter 6, 7) should be systematically examined. The researcher is also invited to accept the challenge made by the authors, since in most chapters and mainly in the framing two chapters (1 and 11) the authors themselves raise questions for further research. For example, in chapter 11, Selden and Selden state when addressing Harel’s chapter (10), “are there ways to help students become better at it [generalizing]? . . . further studies could help provide details that might help teachers engender such reasoning in their students” (p. 216). Furthermore, when addressing Rowland’s chapter (9) they say, “one question to ask from a pedagogical point of view is whether a specific generic proof is likely to be illuminating, even if one can easily find a suitable particular case” (p. 216). They then relate to a number of chapters that discuss students’ conceptions and difficulties, saying that, “it would be an interesting research question to see what sorts of mental images of the division algorithm, or of prime factorization, that university students bring with them and to what degree they are aware of using inner vision or inner speech” (p. 218). Also, “issues regarding divisibility have been raised by various chapter authors, for example, the necessity to distinguish the indivisible units of integer division from the infinitely divisible units of rational division, as well as the importance of negotiating successfully between the modular (2 · 10 + 1), fractional (101 /2 ), and decimal (10.5) representation of ‘21 divided by 2’ . . .. All of these are ripe for further investigation” (p. 220). Indeed, this monograph can be regarded as a necessary and valuable ‘first step’ in the investigation of number theory as a conceptual field, and as a promising field for promoting students’ mathematical reasoning. Still much more interesting research work in this area is needed. In conclusion, this monograph is recommended to mathematics educators and to mathematics education researchers. But even more so, heads or coordinators of mathematics education departments should encourage their teams to read and discuss the various issues highlighted and those merely insinuated in this monograph during department meetings. They may reflect, for instance, on the impressive integration of theory, research and didactical implications found in the monograph. For example, the repeated use of Dubinsky’s APOS theory for the analysis of the participants’ solutions (e.g., in chapter 3), the use of research findings to validate and enlarge the APOS theoretical framework (e.g., in chapter 5), attempts to integrate this theory with another theory to create a new, more suitable theoretical framework (e.g., in chapter 3, 5), or to use the APOS theory in

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designing and ranking the difficulty of number theory tasks (e.g., in chapter 5). In order to enrich these discussions, the department members are invited to add to their reading two very significant extras: Fischbein’s (1993) analysis of students’ formal, algorithmic and intuitive knowledge and Zazkis’s (1999) analysis of students’ reactions to number theory tasks by means of the intuitive rules theory. These may contribute considerably to the interpretation of the data reported about students’ errors and difficulties, about possible reasons that underlie students’ performance, and, perhaps, to novel instructional trends.

R EFERENCES Fischbein, E.: 1993, ‘The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity’, in R. Biehler, R.W. Scholz, R. Straesser and B. Winkelmann (eds.), Didactics of Mathematics as a Scientific Discipline, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 231–245. NCTM [National Council of Teachers of Mathematics]: 1989, Curriculum and Evaluation Standards for School Mathematics, Reston, Virginia. NCTM [National Council of Teachers of Mathematics]: 2000, Principles and Standards for School Mathematics, Reston, Virginia. Zazkis, R.: 1999, ‘Intuitive rules in number theory: Example of “The more of A, the more of B” rule implementation’, Educational Studies in Mathematics 40(2), 197–209. Zazkis, R. and Campbell, S.: 1996a, ‘Divisibility and multiplicative structure of natural numbers: Prospective teachers’ understanding’, Journal for Research in Mathematics Education 27(5), 540–563. Zazkis, R. and Campbell, S.: 1996b, ‘Prime decomposition: Understanding uniqueness’, Journal of Mathematical Behavior 15(2), 207–218.

P ESSIA T SAMIR Dep. of Science Education, School of Education, Tel-Aviv University

BOOK REVIEW TEACHING/LEARNING: FINDING NEW WAYS TO MEANING

Philip C. Clarkson

Perspectives on practice and meaning in mathematics and science classrooms by David Clarke (ed.), Kluwer Academic Publishers. Mathematics Education Library Series, Vol. 25 ISBN 0-7923-6939-4, March 2001, 360 pp. I have not read such an informative book on learning/teaching for a long time. The format that the Editor has used for this volume purposefully ties the whole together. This has been done by exemplifying the theoretical stance that has been taken to try and understand more fully just what is going on to make meaning in mathematics and science classrooms. Now that is not to say there is one way to know what is going on, implying that if only we take the correct, or the most appropriate view for this situation, or a modernist, or Vygotskian approach, or . . . then we will truly ‘know’. Clarke as Editor and as a contributor makes it quite clear time and again that such a goal is just not of interest. Even one classroom session is such a complex set of interactions that to say there is only one way of knowing, observing, understanding that set is facile. There can be many useful questions to ask which demand different approaches. The creative notion at the heart of Clarke’s project developed in this volume is that he did not try to ask such a set of questions himself. He drew together a group of colleagues which together represented a range of views and expertise and asked them to develop their own questions, and appropriate research methodologies for their questions. The common object of study for this group was video data and transcripts of interviews with teachers and students of the same eight secondary mathematics and science lessons. Often when a group is brought together to work on a project, there is normally an underlying assumption that their work will be coherent. That is it will end with some type of consensus. For many of us who have worked in such a manner, although we may have our own tasks, and maybe our own questions or sub questions, we are always looking forward and sideways to try and develop at least an integrated, coherent account of what was happening, in this case in those eight classrooms. The notion of synthesising our contributions or accounts to such an end is simply taken Educational Studies in Mathematics 51: 287–290, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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for granted. But in the project described in this book the emphasis was on ‘complementarity’ rather than ‘consensus’. Although the Editor does in the last chapter synthesise the various sub projects developed in individual chapters, it is made quite clear that this is only one possible ending. The reader is encouraged to be involved and weave their own piece of cloth from the varying approaches and perspectives to gain their own ending. Of course such an approach is easy to encourage. However the Editor and his co researchers are in privileged positions, and this too must be recognised. They clearly have worked as a team over a period of time, and although not reaching for a consensus view, nevertheless have greater insight into each other’s work than can be possibly portrayed in a chapter of a book for the reader. However the crucial, and potentially broadening approach embodied in this volume, is this approach. It seems to me that such an approach does more justice to the layered complexity of classrooms to which some type of understanding is sort. One of the features of each chapter is each team (often of one person) does as they were asked and situates their study quite clearly in a relevant literature. This is in itself a valuable contribution of the volume that will be appreciated by graduate students and researchers. So apart from having accounts of how various approaches can be adopted in analysing a classroom, a useful set of literature for each approach is also available. In other words, there is ample literature cited in the different chapters that will allow the reader to further explore the various ideas used. Most chapters were easy to read and were thought provoking. There is a little variation of style between chapters, but such variation is to be expected if different teams of researchers are to bring their insight to bear on this data set in their own way. Overall I think the quality of the writing ranged from good to excellent. For me the strongest chapters were the first, second and last which sandwiched the individual studies together into a delightful meal. In the first two chapters Clarke introduces the reader to the project and its theorisation, and gives the details of the data set with which the different research teams had to work. In the last he draws together one perspective on how the different views of these lessons found in the intervening chapters can be placed side by side to give some overall feeling of what was happening, and his understanding of advancing our ways of doing classroom research. I found Lerman’s chapter excellent. In this chapter Lerman brings to bear his own depth of knowledge gained from using a Vygotskian framework and brings into being a story of mathematical meaning emphasising the centrality of social relationships. Holton and Thomas also chose to deal with meaning in a mathematical context. Like Lerman they were also

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interested in the situation of a student group since for them it was the language of interaction that offered an interesting lens to understanding. ‘Interest’ for Ainley was the critical construct to use in trying to decipher how meaning was being created in the classroom. In this chapter she nicely shows that getting the students interested is just not enough. Apart from anything, the notion of interest itself is multi faceted and needs to be analysed. Clarke’s and Helme’s following chapter on cognitive engagement seeks to clarify more direct ways to target this phenomena through examination of linguistic and behavioural patterns of students rather than just the indirect measures of time on task and such like. Interestingly they also note that sometimes the presence of the teacher can in fact lower rather than heighten the quality of student inquiry, a fact that would surprise and perhaps upset many mathematics teachers. Rodrigues takes as her pathway into the complexity of the science classroom interactions she explores the notion of ‘context’; in particular the facets of context of communications, resources and processes. She challenges what has perhaps become the norm in science teaching that drawing on ‘everyday’ contexts must some how always be a good thing. She also draws thought provoking contrasts between students-student communication and teacher-student communication, the former being more of a negotiative mode, and the latter dominated by the need to meet the teacher’s expectations. In a chapter that for many may appear to be opening up for the science and mathematics education communities an underresearched area, Reeve and Reynolds zero in on gestures as a way to fruitfully explore classrooms. Doubtless many readers of this book will admit freely that body language has some importance in communication. But these authors suggest that it is more than just a side issue. In particular for some students gestures may be an important medium that they use when they are starting to grasp meaning in complex situations, and it allows them to engage at some level in meaningful discussion. It seems to this reviewer that there is a wealth of opportunities here to explore further. It is hard to point to a weak chapter. However if pressed I found LewisShaw’s chapter a little off the mark in that a very general ‘values’ framework was taken and applied to the quite specific science situation. However since values in classrooms is a topic that is so underresearched and yet so crucial for understanding the multitude of interactions that are played out, any useful study is most welcome. Perhaps Baird’s chapter has a tone bordering on ‘this is the way to go’, when the critical message of the whole book was just the opposite. However his analysis first of learning through a model that encompasses thinking, feeling and acting, then looking at the implications for teaching and deliberately planning for the feeling and

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acting as well as the thinking, using the science classroom as a context is of interest. If this chapter helps researchers to take seriously the notion of planning for what feelings they hope will be invoked in classes, rather than just looking to see what feelings were invoked, I would think this would be an important contribution. One interesting lack in this volume was the missed opportunity of contrasting the mathematics and science classroom situations. Clearly each group of authors had a limit on space in their chapter, and they were asked to situate their studies in a literature that was clear to the reader. This was done and as noted above is a valuable aspect of the volume as a whole. However it seems a pity that for example Ainley’s work revolving around ‘interest’ was confined to science, which at the level of secondary school is probably perceived by students as more interesting per se than mathematics. As well Lewis-Shaw’s work on values also took a science context, whereas values in mathematics classrooms still needs fundamental work of recognising that they are an integral part of interplay of relationships in that context as well. I thought the opportunity to contrast the two situations might have been useful for the reader. However the good thing is that the readers have an opportunity to start exploring such contrasts using a framework modelled on the approach used in this project. This volume makes a comprehensive and original contribution to the field of knowledge of education. I think it will lead to many other research studies and much writing as other authors alternatively take this as a new way of organising their thinking, or in trying to tease out just what is done very well here, and what can be done better with other approaches. Overall I would congratulate the Editor again on a very long project that has produced a volume the profession will read and will, I think, allow us to gain more insight into teaching/learning. P HILIP C. C LARKSON Australian Catholic University Fitzroy, Australia

CONTENTS OF VOLUME 51

Volume 51

Nos. 1–2

2002

Information for authors

1–2

JO BOALER / Exploring the nature of mathematical activity: using theory, research and ‘working hypotheses’ to broaden conceptions of mathematics knowing

3–21

STEPHEN LERMAN, GUORONG XU and ANNA TSATSARONI / Developing theories of mathematics education research: the ESM story 23–40 DEREK FOXMAN and MEINDERT BEISHUIZEN / Mental calculation methods used by 11-year-olds in different attainment bands: a reanalysis of data from the 1987 APU survey in the UK

41–69

BARBARA JAWORSKI / Sensitivity and challenge in university mathematics tutorial teaching

71–94

JEAN-MARIE KRAEMER / Evaluer pour mieux comprendre les enfants et ameliorer sa pratique

95–116

RONNIE KARSENTY / What do adults remember from their high school mathematics? The case of linear functions 117–144 Book Review

145–147

WERNER BLUM / ICMI Study 14: Applications and modelling in mathematics education – Discussion document 149–171 MICHÈLE ARTIGUE / Information about the ICMI Awards

173–174

Instructions for Authors

175–182

Volume 51

No. 3

2002

R.P. BURN / Letter to the editor: Some comments on ‘The role of proof in comprehending and teaching elementary linear algebra’ by F. Uhlig 183–184 JEAN-LUC DORIER, ALINE ROBERT and MARC ROGALSKI / Commentary: Some comments on ‘The role of proof in comprehending and teaching elementary linear algebra’ by F. Uhlig 185–191 CELIA HOYLES and DIETMAR KÜCHEMANN / Students’ understandings of logical implication 193–223 Educational Studies in Mathematics 51: 291–292, 2002.

292 JANE M. WATSON / Inferential reasoning and the influence of cognitive conflict 225–256 LUIS SALDANHA and PATRICK THOMPSON / Conceptions of sample and their relationship to statistical inference 257–270 Book Reviews

271–290

Volume Contents

291–292